Pembrolizumab with 177Lu-PSMA-617 for Prostate Cancer

Inclusion Criteria: 1. Histologically confirmed prostate adenocarcinoma that is progressive metastatic castration-resistant prostate cancer by Prostate Cancer Clinical Trials Working Group 3 (PCWG3) criteria at the time of study entry. 2. Male participants who are at least 18 years of age on the day of signing informed consent. 3. Castrate level of serum testosterone at study entry (\< 50 ng/dL). Note: Participants without prior bilateral orchiectomy are required to remain on Luteinizing hormone-releasing hormone (LHRH) analogue treatment for duration of study. 4. Prior progression on at least one second generation androgen signaling inhibitor including abiraterone, apalutamide, darolutamide, and/or enzalutamide. 5. Adverse events related to prior anti-cancer treatment (excluding LHRH analogs) must have recovered to Grade \<= 1 (except for any grade alopecia and grade \<= 2 neuropathy). 6. Prior radiotherapy is allowed if the last radiotherapy treatment was greater than 2 weeks from start of study treatment on cycle 1, day 1 (C1D1). Note- Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation (\<=2 weeks of radiotherapy) to non-central nervous system (CNS) disease. 7. At least one Prostate-Specific Membrane Antigen (PSMA) Positron Emission Tomography (PET) (PSMA PET) avid lesion on screening PSMA PET. A positive lesion is defined as uptake above background liver. 8. Eastern Cooperative Oncology Group (ECOG) performance status \<= 1 (Karnofsky \>= 70%). 9. Demonstrates adequate organ function as defined below: 1. Adequate bone marrow function: * absolute neutrophil count \>=1,500/microliter (mcL) * platelets \>=100,000/mcL * hemoglobin \> 9.0 g/dL 2. Adequate hepatic function: * total bilirubin \<= 1.5 x upper limit of normal (ULN). In patients with known or suspected Gilbert's disease, direct bilirubin \<= ULN * aspartate aminotransferase (AST)/serum glutamic-oxaloacetic transaminase (SGOT) \<= 2.5 x institutional ULN (\<= 5 x ULN in patients with liver metastases) * alanine aminotransferase (ALT)/serum glutamic-pyruvic transaminase (SGPT) \<= 2.5 x institutional upper limit of normal (\<= 5 x ULN in patients with liver metastases) 3. Adequate renal function: * creatinine \<= 1.5 x within institutional upper limit of normal OR * creatinine clearance Glomerular filtration rate (GFR) \>= 50 mL/min/1.73 m\^2, calculated using the Cockcroft-Gault equation or 24 hour urine collection. 10. Participants must use appropriate methods of contraception during study treatment and for at least 6 months after last study treatment. Patients who are sexually active should consider their female partner to be of childbearing potential if she has experienced menarche and is not postmenopausal (defined as amenorrhea \> 24 consecutive months) or has not undergone successful surgical sterilization. Even women who use contraceptive hormones (oral, implanted, or injected), an intrauterine device, or barrier methods (diaphragms, condoms, spermicide) should be considered to be of childbearing potential. Patients who have undergone vasectomy themselves should also be considered to be of childbearing potential. Acceptable methods of contraception include continuous total abstinence, or double-barrier method of birth control (e.g., condoms used with spermicide, or condoms used with oral contraceptives). Periodic abstinence and withdrawal are not acceptable methods of contraception. 11. Participants must provide consent to comply to recommended radioprotection precautions during study. 12. Participants willing to undergo a tumor biopsy. Bone or soft tissue lesion is allowed. Note: The biopsy can be waived if there is no safely accessible lesion in the judgement of the treating investigator. 13. Participants with previously treated brain metastases are eligible provided the following criteria are all met: 1. Last treatment was \> 28 days prior to C1D1. 2. No evidence of new/progressive brain metastases is observed on magnetic resonance imaging (MRI) obtained during screening window 3. Patient is clinically stable without requirement of steroid treatment for at least 14 days prior to first dose of study treatment on C1D1. 14. Individuals with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. 15. Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: 1. De novo small cell neuroendocrine prostate cancer will not be allowed due to putative lower PSMA expression in this tumor subtype. Note-Treatment-emergent small cell neuroendocrine prostate cancer detected in metastatic tumor biopsy is not excluded. 2. Soft tissue lesions (lymph nodes \> 1.5 centimeter (cm) in short axis, visceral/soft tissue lesions \> 1 cm) on screening Computerized tomography (CT) that are negative on PSMA PET. Note: Negative lesions on PSMA PET are defined as those with uptake below the background liver. 3. Has received other systemic anti-cancer therapies administered within 14 days, or 5 half-lives, whichever is shorter, prior to initiation of study treatment. Note: LHRH analogues are the exception. 4. Untreated brain metastases at study entry. 5. Receipt of prior PSMA-directed treatment (e.g., radiotherapy, immunotherapy, or antibody-drug conjugate). 6. Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-Programmed cell death-ligand 2 (PD-L2) agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, cytotoxic T-lymphocyte-associated protein 4 (CTLA-4), OX 40, CD137). 7. Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to the first dose of study treatment on C1D1. Note: Participants who have entered the follow-up phase of an investigational study may participate as long as it has been 4 weeks after the last dose of the previous investigational agent. 8. Receipt of \> 2 lines of prior taxane-based chemotherapy. 9. Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients. 10. Has an active autoimmune disease that has required systemic treatment in the past 2 years (i.e., with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Note: Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) or treatment with drugs (e.g., neomercazole, carbimazole, etc.) that function to decrease the generation of thyroid hormone by a hyper-functioning thyroid gland (e.g., in Graves' disease) is not considered a form of systemic treatment of an autoimmune disease. 11. Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy at a prednisone equivalent dose of \> 10 mg daily or other form of immunosuppressive therapy within 7 days prior to first dose of study drug. 12. Has a history of (non-infectious) ≥ grade 2 pneumonitis/interstitial lung disease that required steroids within past 2 years or has current ≥ grade 1 pneumonitis/ interstitial lung disease at the time of study enrollment. 13. Has received a live vaccine or live-attenuated vaccine within 30 days prior to the first dose of study drug on C1D1. Note-Administration of a killed vaccine is allowed. 14. Patients who because of age, general medical or psychiatric condition, or physiologic status cannot give valid informed consent. 15. Has clinically significant cardiovascular disease including, but not limited to: 1. Uncontrolled or any New York Heart Association Class 3 or 4 congestive heart failure. 2. Uncontrolled angina, history of myocardial infarction, unstable angina, or stroke within 6 months before study entry. 3. Clinically significant arrhythmias not controlled by medication. Note: Chronic rate controlled, or paroxysmal atrial fibrillation/flutter is not an exclusion to study participation. 16. Prior external beam radiation involving \> 25 percent (%) of bone marrow or within 14 days of start of protocol therapy on C1D1. 17. Major surgery within 28 days of study treatment. Note: If participant received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting study treatment on C1D1. Minor procedures (e.g., biopsy, cataract surgery, stent placement, endoscopy) are not considered major surgery. 18. Has an active infection requiring intravenous antibiotics within 7 days prior to C1D1. 19. Has a known history of Hepatitis B infection (defined as Hepatitis B surface antigen (HBsAg) reactive) or known active Hepatitis C virus infection (defined as (HCV RNA) \[qualitative\] detected, with the following exceptions: 1. Participants who are HbsAg positive are eligible if they have received Hepatitis B virus (HBV) antiviral therapy for at least 4 weeks and have undetectable HBV viral load prior to study entry 2. Participants with history of Hepatitis C virus (HCV) infection are eligible if HCV viral load is undetectable at screening. Participants must have completed curative anti-viral therapy at least 4 weeks prior to study entry. 20. Has a known history of active Bacillus Tuberculosis (TB). 21. Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. 22. History of bleeding diathesis and currently on anti-coagulation therapy that cannot be safely discontinued for the tumor biopsy procedure. 23. Any condition that, in the opinion of the Principal Investigator, would impair the participant's ability to comply with study procedures.