Inclusion Criteria:

• Histological/cytological confirmation of biliary cancer
• Confirmation of advanced biliary cancer that is refractory or intolerant to gemcitabine or fluoropyrimidine based therapy with FGFR2 fusion [using next-gen sequencing assays (such as Foundation One) or fluorescent in situ hybridization (FISH) break-apart assays] or FGFR pathway mutation/amplification [using next-gen sequencing assays (such as Foundation One)]; assays must be performed in a Clinical Laboratory Improvement Amendments [CLIA] certified laboratory and done as a CLIA validated test or research use only [RUO] in a CLIA laboratory
• Measurable disease
• Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
• Absolute neutrophil count (ANC) >= 1500/mm^3
• Platelet count >= 100,000/mm^3
• Hemoglobin >= 9.0 g/dL
• Total bilirubin =< 1.5 x upper limit of normal (ULN), unless due to Gilbert's syndrome
• Aspartate transaminase (AST) and alanine aminotransferase (ALT) < 3 x ULN
• Creatinine =< 1.5 x ULN
• Serum lipase and amylase =< 2.5 x ULN; NOTE: if subject has tumor involvement in the liver =< 3 x ULN
• Negative pregnancy test done =< 7 days prior to registration, for women of childbearing potential only
• Recovered from prior radiotherapy and/or systemic therapy related toxicities to grade =< 1
• Provide informed written consent
• Life expectancy >= 3 months
• Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study); Note: during the Active Monitoring Phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up
• Female and male patients who are fertile agree to use an effective form of contraception with their sexual partners from registration through 4 months after the end of treatment
• Ability to complete questionnaire(s) by themselves or with assistance

Exclusion Criteria:

• Any of the following:
  • Pregnant women
  • Nursing women
  • Men or women of childbearing potential who are unwilling to employ adequate contraception
• 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
• Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive and currently receiving antiretroviral therapy; NOTE: patients with a known history of HIV infection are not eligible for this trial
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
• Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm
• Prior systemic chemotherapy, radiation therapy or major surgery =< 30 days prior to registration
• Concurrent use of any other approved or investigational anticancer agents, including hormonal agents
• Prior nitrosourea or mitomycin C =< 6 weeks prior to registration
• Patients with gastrointestinal comorbidities that would affect intake or absorption of ponatinib
• Untreated or progressive brain metastases
• Prior treatment with or allergic reactions attributed to compounds of similar chemical or biologic composition to ponatinib
• Clinically uncontrolled hypertension (diastolic blood pressure > 90 mm mercury [Hg]; systolic > 140 mm Hg); Note: patients with hypertension should be undergoing treatment at study entry for blood pressure control
• Previous or concurrent malignancy except adequately treated basal or squamous cell skin cancer, in situ carcinoma of the cervix, or other solid tumor treated curatively and without evidence of recurrence for at least 5 years
• History of significant bleeding disorder unrelated to cancer
• History of acute pancreatitis within 1 year prior to registration, chronic pancreatitis, alcohol abuse or uncontrolled hypertriglyceridemia (triglycerides > 450 mg/dL)
• Clinically significant, uncontrolled, or active cardiovascular disease, specifically including, but not restricted to:
  • Any history of myocardial infarction, stroke, or revascularization
  • Unstable angina or transient ischemic attack within 6 months prior to registration
  • Congestive heart failure within 6 months prior to registration, or left ventricular ejection fraction (LVEF) less than lower limit of normal per local institutional standards within 6 months prior to registration
  • History of clinically significant (as determined by the treating physician) atrial arrhythmia
  • Any history of ventricular arrhythmia
  • Active venous thromboembolism including deep venous thrombosis or pulmonary embolism that is not amenable to treatment with anticoagulants
  • Patients with congenital prolonged QT syndromes and abnormal baseline prolonged corrected QT (QTc) (> 450 ms in men and > 470 ms in women)
  • Patients with an ejection fraction =< 50% as assessed by a baseline echocardiogram
• Taking medications that are known to be associated with torsades de pointes
• Taking any medications or herbal supplements that are known to be strong inhibitors of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) =< 14 days prior to registration