Inclusion Criteria:

          -  For Cohorts A, B, C, D, E, G: Has histologically- or cytologically-confirmed
             adenocarcinoma of the prostate without small cell histology

               -  For Cohorts F, H, I: Has t-NE prostate cancer defined by ≥1% neuroendocrine cells
                  in a recent biopsy specimen from a metastasis as determined by the
                  investigational site and confirmed by central histology review prior to
                  enrollment

          -  Is able to provide tumor tissue from a site not previously irradiated as follows:
             Cohorts A, E, and G: must provide a core or excisional biopsy from soft tissue or bone
             biopsy within 1 year of screening and after developing mCRPC; Cohort B: must provide
             an archival tumor tissue sample or tumor tissue from a newly obtained core or
             excisional biopsy from soft tissue if the lesion is clinically accessible; Cohorts C
             and D with soft tissue disease: must provide a core or excisional biopsy from a soft
             tissue lesion if clinically accessible within 1 year of screening and after developing
             mCRPC and an archival specimen if available; and Cohorts F, H, and I must provide a
             core or excisional biopsy from soft tissue or a bone biopsy. Participants with bone
             metastasis only must provide an archival tumor tissue specimen

          -  Has prostate cancer progression within 6 months prior to screening, as determined by
             the investigator, by means of one of the following: PSA progression as defined by a
             minimum of 2 rising PSA levels with an interval of ≥1 week between each assessment
             where the PSA value at screening should be ≥2 ng/mL; radiographic disease progression
             in soft tissue based on Response Evaluation Criteria In Solid Tumors Version 1.1
             criteria with or without PSA progression; radiographic disease progression in bone
             defined as the appearance of 2 or more new bone lesions on bone scan with or without
             PSA progression

          -  Has ongoing androgen deprivation with serum testosterone <50 ng/dL (<2.0 nM).
             Treatment with luteinizing hormone-releasing hormone agonists or antagonists for all
             Cohorts must have been initiated ≥4 weeks prior to first dose of study therapy and
             must be continued throughout the study

          -  Participants receiving bone resorptive therapy (including, but not limited to
             bisphosphonate or receptor activator of nuclear factor kappa-β ligand inhibitor) must
             be on stable doses for ≥4 weeks prior to first dose of study therapy

          -  Women of childbearing potential and male participants must agree to use adequate
             contraception starting with the first dose of study therapy through 120 days after the
             last dose of study therapy for Cohorts A, B, C, D, E, F, G, and H and for 180 days
             after the last dose of chemotherapy for Cohort I

          -  Has a performance status of 0, 1, or 2 on the Eastern Cooperative Oncology Group
             (ECOG) Performance Scale for Cohorts A and C and a performance status of 0 or 1 for
             Cohorts B, D, E, F, G, H, and I within 10 days of study start

          -  For Cohort A: Has received docetaxel for mCRPC. Prior treatment with 1 other
             chemotherapy for mCRPC is allowed. Up to 2 second-generation hormonal manipulations
             (e.g., abiraterone acetate and/or enzalutamide) are allowed. Prior docetaxel for
             metastatic hormone-sensitive prostate cancer is allowed if ≥4 weeks have elapsed from
             the last dose of docetaxel prior to day 1 of Cycle 1

          -  For Cohort B: Has received prior treatment with either abiraterone acetate or
             enzalutamide (but not both) in the prechemotherapy mCRPC state. Participants in Cohort
             B must have received at least 4 weeks of either abiraterone or enzalutamide treatment
             (but not both) who failed treatment or became intolerant of the drug

          -  For Cohort C: Has received prior treatment with abiraterone acetate in the
             pre-chemotherapy mCRPC state without prior enzalutamide. Participants in Cohort C must
             have received at least 4 weeks of abiraterone treatment who failed treatment or become
             intolerant of the drug. Participants who received abiraterone acetate in the
             hormone-sensitive state will not be eligible

          -  For Cohort D: Has not received chemotherapy for mCRPC and has either not had prior
             second generation hormonal manipulation for mCRPC OR has previously been treated with
             enzalutamide for mCRPC and failed treatment or has become intolerant of the drug.
             Prior docetaxel for metastatic hormone-sensitive prostate cancer is allowed if ≥4
             weeks have elapsed from the last dose of docetaxel. Prior treatment with abiraterone
             acetate in the hormone-sensitive metastatic setting is allowed as long as there was no
             progression on this agent and abiraterone acetate was not discontinued due to adverse
             events (AEs)

          -  For Cohorts E, F, and H: Has received docetaxel for mCRPC. Prior treatment with 1
             other chemotherapy for mCRPC is allowed. Up to 2 second-generation hormonal
             manipulations (eg, abiraterone acetate, enzalutamide, apalutamide, darolutamide or
             other next-generation hormonal agents [NHA]) are allowed. Participants who received
             prior ketoconazole for metastatic disease may be enrolled. If docetaxel chemotherapy
             is used more than once (eg, once for metastatic hormone-sensitive and once for mCRPC),
             it will be considered as 1 therapy. Prior docetaxel for metastatic hormone-sensitive
             prostate cancer is allowed if ≥4 weeks have elapsed from the last dose of docetaxel
             prior to Day 1 of Cycle 1

          -  For Cohorts G and I: Has received docetaxel for mCRPC. Prior treatment with 1 other
             chemotherapy for mCRPC is allowed. Up to 2 second-generation hormonal manipulations
             (eg, abiraterone acetate, enzalutamide, apalutamide, darolutamide, or other NHA) are
             allowed. Participants who received prior ketoconazole for metastatic disease may be
             enrolled. If docetaxel chemotherapy is used more than once (eg, once for metastatic
             hormone-sensitive and once for mCRPC), it will be considered as 1 therapy. Prior
             docetaxel for metastatic hormone-sensitive prostate cancer is allowed if ≥4 weeks have
             elapsed from the last dose of docetaxel prior to Day 1 of Cycle 1

          -  For Cohorts H and I: Has aggressive disease progression manifested by progression
             within 6 months of starting next-generation hormonal agents (NHA) for metastatic
             hormone-sensitive prostate cancer (mHSPC) or mCRPC and progression within <6 cycles of
             docetaxel treatment for mCRPC (docetaxel for mHSPC is allowed in addition to docetaxel
             for mCRPC)

        Exclusion Criteria:

          -  Has had a prior anticancer monoclonal antibody (mAb) within 4 weeks prior to first
             dose study therapy or who has not recovered (i.e., Grade ≤1 or at baseline) from AEs
             due to mAbs administered >4 weeks earlier

          -  Has had prior chemotherapy, targeted small molecule therapy abiraterone treatment,
             enzalutamide treatment, or radiation therapy within 2 weeks prior to first dose of
             study therapy or who has not recovered (ie, Grade ≤1 or at baseline) from AEs due to a
             previously administered agent

          -  Is currently participating in and receiving study therapy or has participated in a
             study of an investigational agent and received study drug or used an investigational
             device within 4 weeks of treatment allocation/randomization

          -  Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any
             other form of immunosuppressive therapy within 7 days prior to treatment
             allocation/randomization

          -  Has had a prior radium treatment or treatment with other therapeutic
             radiopharmaceuticals for prostate cancer

          -  Has an active autoimmune disease that has required systemic treatment in past 2 years

          -  Has a history of (non-infectious) pneumonitis/interstitial lung disease that required
             steroids or current pneumonitis/interstitial lung disease

          -  Has previously participated in any other pembrolizumab (MK-3475) trial, or received
             prior therapy with an anti-programmed cell death 1 (anti-PD-1), anti-programmed cell
             death ligand 1 (anti-PD-L1), and anti-programmed cell death ligand 1 (anti-PD-L2)

          -  Has a known history of Human Immunodeficiency Virus (HIV)

          -  Has known active Hepatitis B or Hepatitis C

          -  Has received a live vaccine within 30 days of the first dose of study therapy

          -  Has known active central nervous system metastases and/or carcinomatous meningitis

          -  Has a "superscan" bone scan defined as an intense symmetric activity in the bones and
             diminished renal parenchymal activity on baseline bone scan such that the presence of
             additional metastases in the future could not be evaluated

          -  Has had prior solid, organ or bone marrow transplant

          -  For Cohort A: Has experienced a seizure or seizures within 6 months of study start or
             is currently being treated with cytochrome P450 enzyme (CYP) inducing anti-epileptic
             drugs for seizures

          -  For Cohort A: Is currently receiving strong or moderate inhibitors of CYP3A4 including
             azole antifungals; macrolide antibiotics; or protease inhibitors

          -  For Cohort A: Is currently receiving strong or moderate inducers of CYP3A4

          -  For Cohort A: Has myelodysplastic syndrome

          -  For Cohort A: Has symptomatic congestive heart failure (New York Heart Association
             Class III or IV heart disease), unstable angina pectoris, cardiac arrhythmia, or
             uncontrolled hypertension

          -  For Cohort B: Has received prior treatment with docetaxel or another chemotherapy
             agent for metastatic prostate cancer

          -  For Cohort B: Has peripheral neuropathy Common Terminology Criteria for Adverse Events
             ≥2 except due to trauma

          -  For Cohort B: Has ascites and/or clinically significant pleural effusion

          -  For Cohort B:Has symptomatic congestive heart failure (New York Heart Association
             Class III or IV heart disease)

          -  For Cohort B: Is currently receiving any of the following classes of inhibitors of
             CYP3A4: azole antifungals; macrolide antibiotics; or protease inhibitors

          -  For Cohort C: Has received prior chemotherapy for mCRPC. Prior docetaxel for
             metastatic hormone-sensitive prostate cancer is allowed if ≥4 weeks elapsed from last
             dose of docetaxel. Participants who received abiraterone acetate in the
             hormone-sensitive state will not be eligible

          -  For Cohort C: Has a history of seizure or any condition that may predispose to seizure
             (including, but not limited to prior cerebrovascular accident, transient ischemic
             attack, or brain arteriovenous malformation; or intracranial masses such as a
             schwannoma or meningioma that is causing edema or mass effect)

          -  For Cohort C:Has known or suspected brain metastasis or leptomeningeal carcinomatosis

          -  For Cohort C: Has a history of loss of consciousness within 12 months of the screening
             visit

          -  For Cohort C: Has hypotension (systolic blood pressure <86 millimeters of mercury
             [mmHg]) or uncontrolled hypertension (systolic blood pressure >170 mmHg or diastolic
             blood pressure >105 mmHg) at the screening visit

          -  For Cohort C: Has received treatment with 5-α reductase inhibitors (e.g., finasteride,
             dutasteride), estrogens, and/or cyproterone within 4 weeks prior to Cycle 1

          -  For Cohort C: Has a history of prostate cancer progression on ketoconazole

          -  For Cohort D: Has received prior treatment with docetaxel or another chemotherapy
             agent for metastatic prostate cancer

          -  For Cohort D: Has progressed on prior abiraterone acetate for treatment of castration
             sensitive or resistant metastatic prostate cancer

          -  For Cohort D: Has discontinued prior treatment with abiraterone acetate due to AEs

          -  For Cohort D: Has previously been treated with ketoconazole for prostate cancer for >7
             days

          -  For Cohort D: Has received prior systemic treatment with an azole drug (eg,
             fluconazole, itraconazole) within 4 weeks of Cycle 1, Day 1

          -  For Cohort D: Has uncontrolled hypertension (systolic BP ≥ 160 mm Hg or diastolic BP ≥
             95 mm Hg)

          -  For Cohort D: Has a history of pituitary or adrenal dysfunction

          -  For Cohort D: Has clinically significant heart disease as evidenced by myocardial
             infarction, or arterial thrombotic events in the past 6 months, severe or unstable
             angina, or New York Heart Association Class II-IV heart disease or cardiac ejection
             fraction measurement of <50% at baseline

          -  For Cohort D: Has atrial fibrillation, or other cardiac arrhythmia requiring therapy

          -  For Cohort D: Has a history of chronic liver disease

          -  For Cohort D: Is currently receiving strong CYP3A4 inducers (eg, phenytoin,
             carbamazepine, rifampin, rifabutin, rifapentine, phenobarbital) during abiraterone
             acetate treatment, CYP2D6 substrates with a narrow therapeutic index (for example
             thioridazine), or CYP2C8 substrates with a narrow therapeutic index (for example
             pioglitazone)

          -  For Cohorts E and F: Has a left ventricular ejection fraction (LVEF) below the
             institutional (or local laboratory) normal range, as determined by multigated
             acquisition (MUGA) or echocardiogram (ECHO)

          -  For Cohorts E and F: Has radiographic evidence of encasement or invasion of a major
             blood vessel, or of intratumoral cavitation

          -  For Cohorts E and F: Has prolongation of QT interval by Fredericia (QTcF) interval to
             >480 milliseconds

          -  For Cohorts E and F: Has had major surgery within 3 weeks prior to first dose of study
             interventions

          -  For Cohorts E and F: Has pre-existing ≥Grade 3 gastrointestinal or
             non-gastrointestinal fistula

          -  For Cohorts E and F: Has had significant cardiovascular impairment within 12 months of
             the first dose of study intervention, such as history of New York Heart Association
             >Class II congestive heart failure, unstable angina, myocardial infarction, or
             cerebrovascular accident (CVA)/stroke, cardiac revascularization procedure, or cardiac
             arrhythmia associated with hemodynamic instability

          -  For Cohorts E and F: Has active hemoptysis (bright red blood of at least 0.5 teaspoon)
             within 3 weeks prior to the first dose of lenvatinib

          -  For Cohorts E and F: Has gastrointestinal malabsorption or any other condition that
             might affect the absorption of lenvatinib

          -  For Cohorts G, H, and I: Has had a severe hypersensitivity reaction to treatment with
             another monoclonal antibody

          -  For Cohorts G, H, and I: Has symptomatic ascites or pleural effusion

          -  For Cohort I: Has clinically active diverticulitis, intra-abdominal abscess,
             gastrointestinal obstruction, and/or abdominal carcinomatosis