Binimetinib for People With Relapsed/Refractory BRAF Wild Type Hairy Cell Leukemia and Variant

* INCLUSION CRITERIA: * Histologically confirmed diagnosis of HCL or HCLv according to morphological and immunophenotypic criteria of World Health Organization (WHO) classification \[WHO, 2008 revised 2016\] of lymphoid neoplasm. Participants should have at least one of the following indications for therapy: * Absolute neutrophil count (ANC) \<1 x10\^3/mcL * Hemoglobin \<10g/dL * Platelets\<100 x10\^3/mcL * Symptomatic splenomegaly * Enlarging HCL mass or bone lesion \> 2cm in short axis * Leukemia cell count \>5x10\^3/mcL * Leukemic doubling time \<6 months Participants who have eligible blood counts within 4 weeks prior to initiation of study therapy will not be considered ineligible if subsequent blood counts prior to initiation of study therapy fluctuate and become ineligible up until the time of the initiation of study therapy. * Refractory or relapsed disease - defined as either: * Refractory- no response or disease progression in \<=1 year following first-line treatment with a purine analog, or * Relapsed- having relapsed following treatment with at least 1 prior purine-analog treatments * Participants must be BRAF WT as confirmed from fresh bone marrow aspirate and/or peripheral blood sample, or lymph node/mass by the Laboratory of Pathology (LP), NCI. * Participants who are ineligible for, unable to obtain in a timely manner, cannot access, unwilling to undergo or have failed Moxetumomab Pasudotox trial at NCI. * Age \>=18 years * Eastern Cooperative Oncology Group (ECOG) performance status \<=2 (Karnofsky \>=60%). * Adequate organ and marrow function as defined below: * Total bilirubin \<= 3x upper limit of normal (ULN), unless consistent with Gilbert's (ratio between total and direct bilirubin \> 5) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \<= 5x ULN * Alkaline phosphatase \<=5x ULN * Serum creatinine \<= 1.5 mg/dL or creatinine clearance \>= 60 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal calculated using estimated glomerular filtration rate (eGFR) * Serum albumin \>= 2 g/dL * Prothrombin time (PT)/International Normalized Ratio (INR) \< 2.5x ULN (If on warfarin, PT/INR \< 3.5x ULN; If on any other anticoagulation, Prothrombin time (PT) \< 2.5x ULN * Fibrinogen \>= 0.5x lower limit of normal * The effects of binimetinib on the developing human fetus are unknown therefore participants must use effective methods of contraception as directed below. * Females of childbearing potential (FOCBP) who are sexually active with a non-sterilized male partner must use a highly effective method of contraception and not donate ova prior to study entry and or the duration of study treatment and until 30 days after the last dose of binimetinib. Periodic abstinence, the rhythm method, and the withdrawal method are not acceptable methods of contraception. Females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or those who are premenarchal or postmenopausal (defined as 12 months with no menses without an alternative medical cause). A highly effective method of contraception is defined as one that results in a low failure rate (i.e., less than 1% per year) when used consistently and correctly. Not all methods of contraception are highly effective. Female participants must use a hormonal method in addition to a barrier method alone, to minimize the chance of pregnancy. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Non-sterilized male participants who are sexually active with a female partner of childbearing potential must agree to use methods of contraception that are highly effective or acceptable, and not donate sperm from study entry until 90 days after the last dose of binimetinib. * Breastfeeding participants must be willing to discontinue breastfeeding from study treatment initiation through 3 days after the last dose of the study drug. * Ability of participant to understand and the willingness to sign a written informed consent document. * Must co-enroll in study 10-C-0066: Collection of Human Samples to Study Hairy Cell and other Leukemias, and to Develop Recombinant Immunotoxins for Cancer Treatment EXCLUSION CRITERIA: * Participants who have had chemotherapy, immunotherapy or radiotherapy within 2 weeks prior to the start of study treatment. * Prior therapy with binimetinib. * Participants who are receiving any other investigational agents or have received an investigational agent within 14 days prior to the start of study treatment. * Participants who have undergone major surgery \<= 6 weeks prior to start of study treatment or who have not recovered from side effects of such procedure. * Known hypersensitivity or contraindication to any component of binimetinib or its excipients. * Inability to swallow and retain study drug. * Pregnant women as evaluated by a positive serum or urine beta-human chorionic gonadotropin (beta-hCG) test. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, cardiac disfunction (details as below), uncontrolled pulmonary infection, pulmonary edema or psychiatric illness/social situations that would limit compliance with study requirements. * Evidence of active Hepatitis B Virus (HBV) or Hepatitis C Virus (HCV) infection. Note: Participants with laboratory evidence of cleared HBV or HCV infection may be enrolled. If positive for Hepatitis B core antibody or surface antigen, the participant must be on Tenofovir or Entecavir and Hepatitis B deoxyribonucleic acid (DNA) viral load (VL) must be \<2000 IU/mL * Active second malignancy requiring treatment other than minor resection of indolent cancers like basal cell and squamous skin cancers. * Human immunodeficiency virus (HIV)-posi