Inclusion Criteria:

          -  Patients must have B-cell non-Hodgkin lymphoma or chronic lymphocytic leukemia/small
             lymphocytic lymphoma. Eligible lymphoma subtypes include (but not limited to): mantle
             cell, follicular, lymphoplasmacytic, marginal zone, transformed indolent B cell
             lymphoma (including transformed chronic lymphoid leukemia [CLL]), or diffuse large B
             cell lymphoma that has relapsed after a response to at least one prior therapy regimen
             or is refractory to prior therapy; patients with mantle cell lymphoma must have
             previously been treated with a Bruton tyrosine kinase (BTK) inhibitor and have either
             had disease progression, intolerance, or exposure to the drug for at least 3 months;
             patients with CLL/SLL are eligible if they had disease progression or intolerance to
             BTKis and/or venetoclax; they are also required to have been treated with the other
             agent for at least 3 months (i.e. patients with progression/intolerance to BTKi need
             to be treated with venetoclax for at least 3 months, and patients with
             progression/intolerance to venetoclax need at least 3 months of exposure to a BTKi);
             patients with de novo diffuse large B-cell lymphoma (DLBCL) must meet one of the
             following criteria:

               -  Biopsy-proven refractory disease after a frontline regimen containing both an
                  anthracycline and rituximab or other anti-CD20 antibody (i.e. "primary
                  refractory"), where any disease recurring within 6 months of completion of the
                  regimen is considered refractory

               -  Relapsed or refractory disease after at least one of the following:

                    -  At least 2 lines of therapy (including at least one with an anthracycline
                       and anti-CD20 antibody)

                    -  Autologous stem cell transplant

                    -  Allogeneic stem cell transplant

          -  Patients of any gender, race or ethnicity

          -  Patients must be capable of understanding and providing a written informed consent

          -  Negative serum pregnancy test within 2 weeks before enrollment for women of
             childbearing potential, defined as those who have not been surgically sterilized or
             who have not been free of menses for at least 1 year

          -  Fertile male and female patients must be willing to use an effective contraceptive
             method before, during, and for at least 4 months after the CAR T cell infusion

          -  Patients must have a Karnofsky performance status of >= 60%

          -  Confirmation of diagnosis by internal pathology review of initial or subsequent biopsy
             or other pathologic material at Fred Hutchinson Cancer Research Center (FHCRC)/Seattle
             Cancer Care Alliance (SCCA)/University of Washington (UW)/Harborview Medical Center
             (HMC)

          -  Evidence of CD20 expression by immunohistochemistry or flow cytometry on the tumor
             specimen obtained with the biopsy performed with screening; if the CD20 expression on
             the screening tumor biopsy is unclear or could not be assessed due to technical
             reasons, CD20 expression on a concomitant tumor specimen (such as marrow biopsy or
             circulating tumor cells) may be used to satisfy this requirement

          -  Serum creatinine =< 2.5

          -  Total bilirubin =< 3.0 mg/dL

          -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 5 x the upper
             limit of normal

          -  Adequate pulmonary function, defined as =< grade 1 dyspnea and saturated oxygen (SaO2)
             >= 92% on room air; if pulmonary function test (PFT)s are performed based on the
             clinical judgment of the treating physician, patients with forced expiratory volume in
             1 second (FEV1) >= 50% of predicted and carbon monoxide diffusing capability (DLCO)
             (corrected) of >= 40% of predicted will be eligible

          -  Adequate cardiac function, defined as left ventricular ejection fraction (LVEF) of >=
             50% as assessed by echocardiogram or multigated acquisition (MUGA) scan, or LVEF of
             45-49% and clearance by a cardiologist

          -  Measurable disease that can be accurately measured in at least one dimension as >= 2.0
             cm with computed tomography (CT), ultrasound, or magnetic resonance imaging (MRI)
             techniques; extranodal disease that is measurable by fludeoxyglucose F-18
             (FDG)-positron emission tomography (PET) imaging only will also be allowed; note that
             if an excisional biopsy was performed that removed the sole site of measurable
             disease, the patient will not be eligible for leukapheresis and generation of CAR T
             cell product

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Absence of uncontrolled active infection
             (bacterial, fungal, viral, mycobacterial) not responding to treatment with
             antibiotics, antiviral agents, or antifungal agents

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Absence of active autoimmune disease
             requiring ongoing systemic immunosuppressive therapy

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Negative serum pregnancy test within 2
             weeks before lymphodepletion chemotherapy for women of childbearing potential, defined
             as those who have not been surgically sterilized or who have not been free of menses
             for at least 1 year

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: No treatment with any investigational
             agent on a different clinical trial between enrollment and lymphodepleting
             chemotherapy

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Serum creatinine =< 2.5

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Total bilirubin =< 3.0 mg/dL

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: AST and ALT =< 5 x the upper limit of
             normal

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Adequate pulmonary function, defined as
             =< grade 1 dyspnea and SaO2 >= 92% on room air; if PFTs are performed based on the
             clinical judgment of the treating physician, patients with FEV1 >= 50% of predicted
             and DLCO (corrected) of >= 40% of predicted will be eligible

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Adequate cardiac function, defined as
             left ventricular ejection fraction (LVEF) of >= 50% as assessed by echocardiogram or
             MUGA scan, or LVEF of 45-49% and clearance by a cardiologist; if subject receives
             cardiotoxic chemotherapy after enrollment, repeat echocardiogram or MUGA is required
             to reestablish eligible LVEF

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Patients must have a Karnofsky
             performance status of >= 60%

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Measurable disease that can be
             accurately measured in at least one dimension as >= 2.0 cm with CT, ultrasound, or MRI
             techniques; extranodal disease that is measurable by FDG-PET imaging only will also be
             allowed; note that if an excisional biopsy was performed that removed the sole site of
             measurable disease, the patient is not be eligible for lymphodepletion and CAR T cell
             infusion; measurable disease can be based on the imaging study done during the
             screening unless the patient received treatment in the interim, in which case imaging
             should be repeated

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Patients must require no corticosteroid
             therapy or dose of less than 15 mg per day of prednisone or the equivalent; pulsed
             corticosteroid dose for disease control is acceptable until the day before the start
             of lymphodepletion

          -  ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Patients must have no active acute or
             chronic GVHD

        Exclusion Criteria:

          -  Active autoimmune disease requiring systemic immunosuppressive therapy

          -  Patients requiring corticosteroid therapy at a dose of 15 mg or more per day of
             prednisone or the equivalent; pulsed corticosteroid dose for disease control is
             acceptable

          -  Patients who are human immunodeficiency virus (HIV) seropositive

          -  Women who are pregnant or breastfeeding

          -  Significant cardiovascular diseases within the past 6 months including uncontrolled
             congestive heart failure (> New York Heart Association [NYHA] class II), myocardial
             infarction, unstable angina, or uncontrolled arrhythmia

          -  History of severe immediate hypersensitivity reaction to cyclophosphamide or
             fludarabine

          -  History or presence of clinically relevant non-lymphoma central nervous system
             pathology, including seizures that are uncontrolled on anticonvulsant therapy (>= 1
             seizure in the last year), paresis, aphasia, stroke, severe brain injuries, dementia,
             Parkinson disease, cerebellar disease, or psychosis

          -  Treatment with any investigational agent on a different clinical trial within 4 weeks
             prior to enrollment, unless the patient is documented to be unresponsive to the
             therapy and at least 3 half-lives have elapsed prior to enrollment

          -  Treatment with any anti-CD19 or anti-CD20 antibody or antibody-drug conjugate therapy
             within 4 weeks before enrollment

          -  Previous treatment with CD19-targeted CAR T cells that has resulted in ongoing B cell
             aplasia at the time of enrollment; patients that demonstrate recovery of normal B
             cells (>= 20 B cells/ul) by flow cytometry at any point 28 days or later after CD19
             CAR T cell infusion will be considered to have functional loss of CD19 CAR T cells and
             are potentially eligible

          -  Known active central nervous system metastases and/or lymphomatous meningitis;
             patients with isolated cerebrospinal fluid (CSF) involvement detectable by flow
             cytometry are eligible if clinically asymptomatic and if abnormal B cells are reported
             to be less than 3% by flow cytometry; subjects with previously treated central nervous
             system (CNS) disease may participate provided: 1) any CNS-directed treatment was
             completed at least 1 month prior to enrollment, 2) imaging studies and CSF evaluation
             show no evidence of disease progression, and 3) any neurologic symptoms have returned
             to baseline

          -  Presence of active acute or chronic graft versus host disease (GVHD)

          -  Uncontrolled active infection (bacterial, fungal, viral, mycobacterial) not responding
             to treatment with intravenous antibiotics, antiviral or antifungal agents

          -  Patients with concurrent known additional malignancy that is progressing and/or
             requires active treatment; exceptions include squamous or basal cell carcinoma of the
             skin and low grade prostate carcinoma (Gleason grade =< 6)

          -  Patients with blood or platelet transfusion within 1 week prior to signing Consent A,
             or with platelets < 50,000/mm^3, neutrophils < 750/mm^3, or hemoglobin < 8.5 g/dL,
             unless the cytopenias are considered by the treating physician to be largely due to
             marrow involvement by lymphoma