Enfortumab Vedotin for the Treatment of Patients With Metastatic or Unresectable Squamous Cell Carcinoma of the Penis

Inclusion Criteria: * Age ≥ 18 years * Histological confirmation of squamous cell carcinoma of the penis (PSCC): NOTE: Biopsy confirmation of at least one site of metastasis is encouraged but not required. * At least one site of metastatic or unresectable PSCC. NOTE: Prior therapy is not required for patients whose treatment is considered palliative (for example, presence of distant metastasis). NOTE: Patients who are potentially curable (any T, N1 - N3, M0) must have had tumor progression after standard chemotherapy, radiotherapy, or surgery, or be unable to receive such treatment. Eligible stages include: * Any T, N1 (i.e., a palpable mobile unilateral inguinal lymph node), M0 OR * Any T, N2 (i.e., palpable mobile multiple or bilateral inguinal lymph nodes), M0 OR * Any T, N3 (i.e., fixed inguinal nodal mass or any pelvic lymphadenopathy), M0 OR * Any T, any N, M1 * Patients with clinical N1, M0 mPSCC at protocol entry must be ineligible for surgery because of comorbidities or clinical T4 disease, or have refused surgery * Patients with clinical N1 - N3, M0, and no prior systemic therapy must be: * Unable to receive neoadjuvant (paclitaxel + ifosfamide + cisplatin) TIP because of comorbidities or refused TIP; AND * Unable to receive radiotherapy with curative intent, or refused radiotherapy * Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST 1.1) * Prior therapy is allowed. Patients may be treatment-naïve or have had any number of prior anti-cancer treatments * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Hemoglobin ≥ 9.0 g/dL obtained ≤15 days prior to registration * Absolute neutrophil count (ANC) ≥ 1000/mm\^3 obtained ≤ 15 days prior to registration * Platelet count ≥ 100,000/mm\^3 obtained ≤ 15 days prior to registration * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) or ≤ 3 x ULN for patients with Gilbert's disease obtained ≤ 15 days prior to registration * Alanine aminotransferase (ALT) and aspartate transaminase (AST) ≤ 2.5 x ULN obtained ≤ 15 days prior to registration * Glomerular filtration rate (GFR) or calculated creatinine clearance ≥ 30 ml/min as estimated using the Cockcroft-Gault formula or as measured by 24-hour urine collection obtained ≤ 15 days prior to registration * Provide written informed consent * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) Exclusion Criteria: * Pure verrucous carcinoma of the penis * Non-squamous malignancy of the penis * Squamous carcinoma of the urethra * Preexisting sensory or motor neuropathy ≥ grade 2 * Active central nervous system (CNS) metastases. Exception: Treated CNS metastases are allowed if all of the following are true: * CNS metastases are clinically stable for ≥ 6 weeks prior to registration * If needed, steroid dose is stable and ≤ 20 mg/day of prednisone or equivalent for ≥ 2 weeks prior to registration * Baseline imaging shows no evidence of new or enlarged brain metastasis * No leptomeningeal disease * History of uncontrolled diabetes mellitus ≤ 3 months prior to registration NOTE: Uncontrolled diabetes is defined as hemoglobin A1c (HbA1c) ≥ 8.0% or HbA1c 7.0-7.9% with associated diabetes symptoms (polyuria or polydipsia) that are not otherwise explained * Failure to recover from any of the following therapies prior to registration: * Major surgery * Radiotherapy, chemotherapy, biologics, investigational agents, and/or antitumor treatment with immunotherapy * Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive * Uncontrolled intercurrent illness including, but not limited to: * Ongoing or active infection requiring systemic treatment * History of cerebral vascular event (stroke or transient ischemic attack) * Myocardial infarction or symptomatic congestive heart failure (New York Heart Association \[NYHA\] Class III-IV) ≤ 6 months prior to registration * Unstable angina pectoris * Cardiac arrhythmia * Or psychiatric illness/social situations that would limit compliance with study requirements (e.g., history of substance abuse) * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Currently receiving systemic antimicrobial treatment for viral, bacterial or fungal infection. NOTE: Routine antimicrobial prophylaxis is allowed * Known active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or active hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid (RNA) \[qualitative\] is detected) * Known active keratitis or corneal ulcerations. NOTE: Superficial punctate keratitis is allowed if the disorder is being adequately treated * Known hypersensitivity to enfortumab vedotin or to any excipient contained in the drug formulation of enfortumab vedotin (including histidine, trehalose dihydrate and polysorbate 20) OR subject has known hypersensitivity to biopharmaceutical produced in Chinese hamster ovary cells * Other active malignancy ≤ 2 years prior to registration. EXCEPTIONS: Locally curable cancers that have been apparently cured such as basal or squamous cell skin cancer, non-muscle-invasive bladder cancer, or carcinoma in situ of the breast or low risk Gleason 6 prostate cancer. * History of myocardial infarction ≤ 6 months, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Patients who are sexually active and unwilling to use effective contraception (if they are not already surgically sterile) * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Chemotherapy-naïve patients who are potentially curable (any T, N1 - N3, M0) in the absence of any condition that precludes cisplatin-based chemotherapy, such as low GFR, peripheral neuropathy, hearing impairment, or psychosocial considerations