Inclusion Criteria:

        Patients must meet the following criteria for study entry:

          1. Patient who are at least 18 years of age who have provided written informed consent.

          2. Histologically confirmed adenocarcinoma of the prostate without neuroendocrine or
             small cell differentiation.

          3. Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 1 (see Appendix 1).

          4. Patients must have progressed on prior enzalutamide, abiraterone and/or apalutamide
             for treatment of prostate cancer.

          5. Determination of disease progression on second generation androgen receptor targeted
             agent determined by the local investigator. Progressive disease is defined by PCWG3 as
             any one of the following:

               -  PSA progression: minimum of two rising PSA values from a baseline measurement
                  with an interval of ≥ 1 week between each measurement. The PSA value at screening
                  should be ≥ 1ng/ml.

               -  Soft tissue or visceral disease progression as per modified RECIST 1.1 criteria
                  (see Appendix 2)

               -  Bone progression: ≥ 2 new lesions on bone scan (Appendix 2)

          6. At least 2 weeks since the completion of surgery or radiotherapy prior to
             registration. Any clinically relevant sequelae from the surgery or radiotherapy must
             have improved to grade 1 prior to registration.

          7. Prior surgical orchiectomy or chemical castration maintained on luteinizing
             hormone-releasing hormone (LHRH) analog (agonist or antagonist). Patients without
             prior surgical castration must be currently taking and willing to continue luteinizing
             hormone-releasing hormone (LHRH) analog (agonist or antagonist) therapy throughout the
             duration of study treatment.

          8. Serum testosterone levels ≤ 50ng/dL. (≤ 1.75nmol/L) within 28 days before
             registration.

          9. Imaging evidence of metastatic disease documented with either bone scan or CT scan
             (Appendix 2).

         10. Prior prostate cancer vaccine therapy, radiation therapy, systemic therapies,
             diethylstilboestrol (DES) or other estrogens are allowed up to 28 days prior to trial
             registration. Note: bicalutamide flutamide or nilutamide must be discontinued within 4
             weeks of registration.

         11. Significant PSMA avidity on 68Ga/18F-PSMA PET/CT, defined as a minimum uptake of
             SUVmax 20 at a site of disease, and SUVmax > 10 at other sites of disease ≥10mm
             (unless subject to factors explaining a lower uptake, e.g. respiratory motion,
             reconstruction artefact).

         12. Patients must have a life expectancy ≥ 24 weeks.

         13. Patients must agree to use a highly effective form of contraception for the entire
             duration of the study plus an additional 120 days (a spermatogenesis cycle) after the
             last dose of study treatment and refrain from donating sperm during this period (see
             section 10.3.3).

         14. Patients must be willing and able to comply with the protocol for the duration of the
             study including undergoing treatment and scheduled assessments.

         15. Patients must have adequate bone marrow, hepatic and renal function documented within
             28 days of registration, defined as:

               -  Haemoglobin ≥90 g/L independent of transfusions (no red blood cell transfusion in
                  last 4 weeks)

               -  White blood cells >3x109/L

               -  Absolute neutrophil count ≥1.5x109/L

               -  Platelets ≥100 x109/L

               -  Total bilirubin ≤1.5 x upper limit of normal (ULN) except for patients with known
                  Gilbert's syndrome.

               -  Aspartate transaminase (AST) (SGOT) and alanine transaminase (ALT) (SGPT) ≤2.5 ×
                  ULN or ≤5 × ULN for participants with liver metastases.

               -  Serum creatinine ≤1.5 x ULN or a calculated creatinine clearance > 50mL/min
                  (Chronic Kidney Disease Epidemiology (CKD-EPI) equation for patients with
                  creatinine levels above institutional normal.

               -  Albumin >30 g/dl

               -  International normalized ratio (INR), prothrombin time (PT), activated partial
                  thromboplastin time (aPTT) ≤1.5 × ULN unless participant is receiving
                  anticoagulant therapy as long as PT or aPTT is within therapeutic range of
                  intended use of anticoagulants.

         16. Patients who are deemed by PSMA imaging to have readily accessible disease will be
             required to consent to 3 serial tumour biopsies - at screening, post combination
             treatment (at any time between weeks 2-4) and on progression

        Exclusion Criteria:

          1. Site(s) of disease that are FDG positive with low PSMA expression defined by PSMA
             SUVmax < 10.

          2. Previous history or presence of brain metastases or leptomeningeal metastases.

          3. Any prior exposure to anti-PD-1, anti-PD-L1/L2, anti-CTLA-4 antibody or any other
             antibody or drug specifically targeting T cell co-stimulation or checkpoint pathway.

          4. Any prior treatment with cabazitaxel.

          5. Any prior exposure to 177Lu-PSMA.

          6. Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements.

          7. Patients with active, known or suspected autoimmune disease including Sjogren's
             syndrome. Patients with vitiligo, type I diabetes mellitus, residual hypothyroidism
             due to autoimmune condition only requiring hormone replacement, psoriasis not
             requiring systemic treatment, or conditions not expected to recur in the absence of an
             external trigger are eligible.

          8. Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients.

          9. Participants must have recovered from all AEs due to previous therapies to ≤Grade 1 or
             baseline. Participants with ≤ Grade 2 neuropathy may be eligible.

         10. Patients with a condition requiring systemic treatment with either corticosteroids (>
             10mg daily prednisone equivalents) or other immunosuppressive medications within 14
             days of registration. Inhaled or topical steroids, and adrenal replacement doses ≤ 10
             mg daily prednisone equivalents are permitted in the absence of active autoimmune
             disease.

         11. Other malignancies within the previous 2-years other than, melanoma in situ, basal
             cell or squamous cell carcinomas of skin with a > 30% probability of recurrence within
             12 months.

         12. Positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus
             ribonucleic acid (HCV antibody) indicating acute or chronic infection.

         13. Patient has a known history of Human Immunodeficiency Virus (HIV).

         14. Patients with symptomatic or impending cord compression unless appropriately treated
             beforehand and clinically stable for ≥ 4 weeks.

         15. Previous history of interstitial lung disease or non-infectious pneumonitis.

         16. Recent administration of a live vaccine within 30 days prior to the first dose of
             study drug. Examples of live vaccines include, but are not limited to, the following:
             measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies,
             Bacillus Calmette-Guérin (BCG), and typhoid vaccine. Seasonal influenza vaccines for
             injection are generally killed virus vaccines and are allowed; however, intranasal
             influenza vaccines (eg, FluMist®) are live attenuated vaccines and are not allowed.

         17. Recent administration of the influenza vaccine (within 30 days of registration).