A Study of Tarlatamab for People With Prostate Cancer

Inclusion Criteria: * To be included in this study, participants should complete all screening procedures and meet all of the following criteria: * Willing and able to provide, or have a legally authorized representative provide, written informed consent and privacy authorization for the release of personal health information. A signed informed consent must be obtained before screening procedures are performed. NOTE: Privacy authorization may be either included in the informed consent or obtained separately. * 18 years of age and above * Resting oxygen saturation of ≥ 90% on room air. * Histologically confirmed prostate cancer. Any histologic subtype of prostate cancer is allowed. * Documented metastatic disease based on conventional imaging (soft tissue disease on computed topography (CT)/magnetic resonance imaging (MRI), or at least 2 lesions as found on bone scan) obtained during Screening. Metastatic disease as seen only on PET scan is exclusionary. Metastatic disease may include pelvic lymph nodes above and/or below the aortic bifurcation. Note: Measurable disease by RECIST 1.1 criteria is not required; however, a minimum of 50% of participants enrolled must have measurable disease by RECIST 1.1 criteria. * Serum testosterone ≤ 50 ng/dL with ongoing androgen-deprivation therapy (ADT) or de novo small cell NEPC (neither testosterone levels nor ADT are required in participants with de novo small cell NEPC). * Progression on at least one line of therapy in the metastatic setting based on at least one of the following criteria: 1. Prostate-specific antigen (PSA) progression defined as a minimum of 2 rising PSA levels with a minimum of a 1-week interval between each determination. There is no minimum PSA level required. 2. Nodal or visceral progression as defined by RECIST 1.1 with PCWG3 modifications 3. Progression of bone disease with two or more new bone lesions on bone scan (i.e., PCWG3) * Participants with de novo small cell NEPC are required to have received prior platinum-based chemotherapy or be ineligible for this treatment. * No more than two prior lines of cytotoxic chemotherapy in the metastatic castration-resistant disease setting or the de novo small cell NEPC setting * DLL3 positive disease as defined by archival or fresh tumor biopsy with positive DLL3 expression using a CLIA certified assay (50% or more of tumor cells with DLL3 expression by IHC). DLL3 testing may be obtained at any point prior to study enrollment * Participants with brain metastases are eligible provided definitive treatment completed at least two weeks prior to C1D1, no concurrent steroids for the treatment of central nervous system (CNS) disease, and no progression noted on CNS imaging obtained during screening obtained following completion of definitive treatment. * ECOG status of ≤ 2 * Normal organ function with acceptable initial laboratory values within 14 days of treatment start. Red blood cell transfusions during screening may be allowed if laboratory values initially fall outside of the following ranges: * Absolute neutrophil count (ANC) ≥ 1,500/μ * Hemoglobin ≥9g/dL * Platelet count ≥75,000/μl * Bilirubin ≤ 1.5 upper limit of normal (ULN) or \< 2 if liver metastases or Gilbert's disease * SGOT (AST) \< 3 x ULN or \< 5 if liver metastases * SGPT (ALT) \< 3 x ULN or \< 5 if liver metastases * Adequate renal function CrCl ≥ 30mL/min using Cockroft Gault or MDRD calculation * Participants must agree to use a medically acceptable method of birth control (e.g., spermicide in conjunction with a barrier such as a condom) or sexual abstinence for the duration of the study, including 60 days after the last dose of study drug. Sperm donation is prohibited during the study and for 60 days after the last dose of study drug. Female partners must use hormonal or barrier contraception unless postmenopausal or abstinent. Exclusion Criteria: * Any prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessments in the judgment of the site Principal Investigator (PI). * Medical conditions such as uncontrolled hypertension (sustained SBP \> 160 mm Hg or diastolic BP \> 100 mm Hg), any history of seizure, or major cardiovascular event (myocardial infarction), stroke or transient ischemia event within 6 months prior to study entry, or NYHA class ≥ III congestive heart failure, or uncontrolled cardiac arrhythmia. * Active hepatitis B or C infection (defined as positive HBsAg or positive HBV DNA in participants who are HBV core Ab +; detectable HCV RNA by PCR). Prior treatment for HBV or HCV is allowed. * Active human immunodeficiency (HIV) infection on antiviral therapy as measured by a detectable viral load. * History of leptomeningeal disease. * Active autoimmune disease requiring systemic treatment within the past 2 years or Grade \> 2 autoimmune adverse effect from prior immune checkpoint inhibition (exception: any grade endocrine disorders on replacement treatment are allowed). Prednisone or equivalent at doses of up to 10 mg/day along with oral weekly methotrexate are allowed. No other immunosuppressive medications are allowed * History of interstitial lung disease and/or Grade ≥ 2 pneumonitis at the time of study entry * Diagnosis of immunodeficiency or receiving systemic steroid therapy (prednisone \> 10 mg/day or equivalent) within 7 days of C1D1 * Presence of infection requiring IV antibiotics within 7 days of C1D1 * Prior DLL3-targeting treatment * Systemic anti-cancer treatment (other than LHRH analog) within 14 days or 5 half-lives, whichever is shorter, prior to C1D1 * Receipt of another investigational therapeutic agent within 14 days or 5 half-lives, whichever is shorter, prior to C1D1 * Major surgical procedure within 28 days prior to C1D1 * Palliative radiotherapy if \< 1 week prior to C1D1 * Use of any prohibited concomitant medications (Appendix C: Medications With the Potential for Drug-Drug Interactions) within two weeks prior to C1D1. * Grade \> 2 treatment-related adverse event related to prior therapy that is ongoing at the start of study treatment. * Known allergy to any of the compounds under investigation * Any other condition which, in the opinion of the Investigator, would preclude participation in this trial