Phase I Study of Memory-Like Natural Killer Cells With Nivolumab and Relatlimab in Advanced or Metastatic Melanoma After Progression on Checkpoint Inhibitors

Inclusion Criteria: * Diagnosis of histologically confirmed advanced or metastatic melanoma that has progressed after at least 12 weeks or a minimum of 2 doses of treatment with a standard of care PD1/PDL1 containing therapy (nivolumab, pembrolizumab, atezolizumab, or durvalumab). * Age: ≥18 years of age * Have an Eastern Cooperative Oncology Group Performance Status (ECOG) ≤ 2 at screening Form Arm 1 only: Patients must meet the eligibility criteria to undergo apheresis to obtain autlogous NK cells. * For Arm 2 only: Patient must have an available allogeneic NK cell donor who meets the eligibility criteria. * Adequate organ function as defined below: * Total bilirubin \< 2 mg/dL * AST(SGOT)/ALT(SGPT) \< 3.0 x ULN * Creatinine within normal institutional limits OR creatinine clearance \> 40 mL/min/1.73 m\^2 by Cockcroft-Gault Formula * Oxygen saturation ≥ 90% on room air * Ejection fraction ≥ 45% * Patients with a prior history of symptomatic CNS metastases must have received treatment and be neurologically stable for at least for 4 weeks and off anti-seizure medication and steroids for 7 days prior to initiation of LDC. * Able to be off corticosteroids and any other immune suppressive medications for at least 14 days prior to apheresis or lymphodepletion and continuing until 30 days after the infusion of the ML NK cells. However, use of physiological dosing of corticosteroids (defined as ≤15mg prednisone or equivalent) is permitted if deemed medically necessary. * Women of childbearing potential must have a negative pregnancy test within 28 days prior to study registration. Female and male patients (along with their female partners) must agree to use two forms of acceptable contraception, including one barrier method, throughout participation in the study and for at least 5 months after the last dose of relatlimab. * Life expectancy \>12 weeks * Ability to understand and willingness to sign an IRB approved written informed consent document Exclusion Criteria: * Active autoimmune disorder requiring immunosuppression (physiologic steroids defined as ≤15mg prednisone or equivalent are acceptable). * Prior history of an immune-related Grade 3 or 4 AE attributed to prior cancer immunotherapy (other than endocrinopathy managed with either replacement therapy or asymptomatic elevation of serum amylase or lipase) that resulted in permanent discontinuation of the prior immunotherapeutic agent. * Patients with Grade ≤2 irAE who have not completely recovered from irAE (i.e. have residual toxicities \>Grade 1) related to prior cancer immunotherapy (other than endocrinopathy management with replacement therapy or stable vitiligo). Patients treated with corticosteroids for irAE must demonstrate absence of related signs or symptoms for ≥7 days following discontinuation of corticosteroids. * Leptomeningeal disease, carcinomatous meningitis, or symptomatic CNS metastases. Patients with asymptomatic brain metastasis with no pending intervention needed, or patients with treated CNS disease and stable for at least 4 weeks and off anti-seizure medication and steroids for 7 days prior to initiation of LDC are eligible. * Known hypersensitivity to one or more of the study agents. * Comorbidities and any conditions, that in the opinion of the investigator, that put the subject at unacceptable risk for study therapy or prevent the participant from consenting or participating in the study. * Uncontrolled and active systemic infections, including but not limited to HIV, Hepatitis B or C infection. * Uncontrolled angina, severe uncontrolled ventricular arrhythmias, or EKG suggestive of acute ischemia or active conduction system abnormalities. * New progressive pulmonary infiltrates concerning for new or uncontrolled infectious process. Infiltrates attributed to infection must be stable/ improving after 1 week of appropriate therapy (4 weeks for presumed or proven fungal infections). * Received any investigational or off-label drugs, or cytotoxic chemotherapy within the 14 days or five half-lives (whichever is greater) prior to apheresis. * Pregnant or breastfeeding. * Subjects are not acceptable candidates if they received prior tumor infiltrating lymphocytes (TIL) therapy (either in the setting of clinical trial or standard of care if TIL therapy is FDA approved in the future), or an organ allograft. * Has a known additional malignancy that is progressing or required active treatment within the past 2 years. Note: participants with basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or carcinoma in situ (eg. Breast carcinoma, cervical cancer in situ) that has undergone potentially curative therapy are not excluded. * Received a live or attenuated vaccine within 28 days prior to the beginning of the lymphodepletion therapy. Eligibility Criteria for Haploidentical Donors (For Arm 2 only) * Donor must be at least 18 years of age. * Donor must be willing, in general good health, and medically able to tolerate leukapheresis required for harvesting the NK cells for this study. * Donor must be negative for hepatitis, HTLV, and HIV on donor viral screen. * Donor may not be pregnant and/or breastfeeding. Women of childbearing potential must have a negative pregnancy test within 30 days prior to apheresis. * Donor must be able to understand and willing to sign an IRB-approved written informed consent document. * Only haploidentical donors will be included. * Donor must meet the requirements of institutional donor guidelines, including the requirements of Foundation for the Accreditation of Hematopoietic Cell Therapy (FACT) criteria. Eligibility Criteria for Autologous Patients (For Arm 1 only) * Patient must be willing and medically able to tolerate leukapheresis required for harvesting the NK cells for this study. * Patient must be negative for hepatitis, HTLV, and HIV on the viral screen. * Patient may not be treated with any cytotoxic treatment within 2 weeks prior to leukapheresis.