Inclusion Criteria:

          -  Participants must have histologically or cytological confirmed stage IIIB/IV NSCLC

          -  Documented ALK-rearrangement (or ROS1- rearrangement) break-apart fluorescence in situ
             hybridization (FISH) (in >= 15% or tumor cells), or next generation sequencing assay
             performed on tumor sample or cell free DNA in a Clinical Laboratory Improvement Act
             (CLIA)-approved laboratory

          -  At least one prior ALK or ROS1 targeted tyrosine kinase inhibitor (TKI). With
             progression or intolerance of most recent regimen

          -  Dose Escalation Phase Only: At least one prior ALK or ROS1 targeted TKI. With
             progression or intolerance of most recent regimen.

          -  Dose Expansion Phase Only: In addition to the criteria in Exclusion Criteria # 3, ALK+
             patients with no prior chemotherapy, immunotherapy, radiation therapy or other
             systemic therapy are also allowed.

          -  Measurable or evaluable disease defined by Response Evaluation Criteria in Solid
             Tumors (RECIST) 1.1 criteria

          -  Age >= 18 years

          -  Life expectancy of at least 3 months

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 0-2

          -  Leukocytes >= 3,000/microliter (mcL)

          -  Absolute neutrophil count >=1,500/mcL

          -  Platelets >= 75,000/mcL

          -  Hemoglobin (Hgb) >= 9 gm/dL

          -  Total bilirubin within normal institutional limits

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase (SGOT)) =< 5
             x institutional upper limit of normal

          -  Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase (SGPT)) =< 5 x
             institutional upper limit of normal if there are liver metastases

          -  Creatinine within normal institutional limits OR

          -  Creatinine clearance >= 50 mL/min/1.73 m^2 for participants with creatinine levels
             above institutional normal

          -  Female participants who:

               -  Are postmenopausal for at least 1 year before the screening visit, OR

               -  Are surgically sterile, OR

               -  If they are of childbearing potential, agree to practice 2 effective methods of
                  contraception, at the same time, from the time of signing the informed consent
                  through 4 months after the last dose of the study drugs, or

               -  Agree to completely abstain from heterosexual intercourse

          -  Male participants, even if surgically sterilized (i.e., status post-vasectomy), who:

               -  Agree to practice effective barrier contraception during the entire study
                  treatment period and through 4 months after the last dose of the study drugs, or

               -  Agree to completely abstain from heterosexual intercourse

          -  Negative pregnancy testing required at screening and cycle 1 day 1 for women of
             childbearing potential

          -  Ability to understand a written informed consent document, and willingness to sign it

          -  Able to swallow and retain orally administered medication and does not have any
             clinically significant gastrointestinal abnormalities that may alter absorption such
             as malabsorption syndrome or major resection of the stomach or bowels

          -  Participant is deemed by the investigator to have the initiative and means to be
             compliant with the protocol (treatment and follow-up)

        Exclusion Criteria:

          -  Is currently participating and receiving study therapy or has participated in a study
             of an investigational agent and received study therapy or used an investigational
             device within 7 days for prior TKI and 14 days for chemotherapy or radiation and 30
             days for prior immunotherapy before the first dose of treatment

          -  Prior treatment with either study drug as most recent treatment or prior
             discontinuation of either study drug due to toxicity

          -  Known hypersensitivity to any study drug components

          -  Known history of interstitial lung disease or interstitial pneumonitis, including
             clinically significant radiation pneumonitis

          -  Known history of myositis

          -  History of pancreatitis

          -  History or current evidence of retinal vein occlusion (RVO) or current risk factors to
             RVO (e.g. uncontrolled glaucoma or ocular hypertension, history of hyperviscosity or
             hypercoagulability syndromes); history of retinal degenerative disease

          -  Impaired cardiovascular function or clinically significant cardiovascular diseases,
             including any of the following:

               -  Symptomatic chronic heart failure grade >= 2, history or current evidence of
                  clinically significant cardiac arrhythmia and/or conduction abnormality < 6
                  months prior to screening except atrial fibrillation and paroxysmal
                  supraventricular tachycardia.

               -  Have significant, uncontrolled, or active cardiovascular disease, specifically
                  including, but not restricted to:

                    -  Myocardial infarction (MI) within 6 months prior to the first dose of
                       brigatinib

                    -  Unstable angina within 6 months prior to first dose

                    -  Symptomatic congestive heart failure requiring treatment (New York Heart
                       Association grade >= 2), history or current evidence of clinically
                       significant cardiac arrhythmia and/or conduction abnormality =< 6 months
                       prior to Screening except atrial fibrillation and paroxysmal
                       supraventricular tachycardia

                    -  Congestive heart failure requiring treatment (New York Heart Association
                       grade >= 2)

                    -  Left ventricular ejection fraction (LVEF) < 50% as determined by multigated
                       acquisition scan (MUGA) or echocardiogram (ECHO)

                    -  QT interval corrected for heart rate using the Fridericia formula (QTcF) >
                       480 msec;

                    -  History of clinically significant atrial arrhythmia

                    -  Any history of ventricular arrhythmia

                    -  Cerebrovascular accident or transient ischemic attack within 6 months prior
                       to first dose

          -  Uncontrolled hypertension defined as persistent elevation of systolic blood pressure
             >= 150 mmHg or diastolic blood pressure >= 100 mmHg despite current therapy

          -  History of chronic inflammatory bowel disease or Crohn?s disease requiring medical
             intervention (immunomodulatory or immunosuppressive medications or surgery) =< 12
             months prior to first dose of study treatment

          -  Impaired gastrointestinal (GI) function or disease that may significantly alter the
             absorption of binimetinib or brigatinib (e.g., ulcerative diseases, uncontrolled
             vomiting, malabsorption syndrome, small bowel resection with decreased intestinal
             absorption)

          -  History of thromboembolic or cerebrovascular events =< 6 months prior to starting
             study treatment, including transient ischemic attacks, cerebrovascular accidents, deep
             vein thrombosis or pulmonary emboli

          -  Current neuromuscular disorder(s) associated with elevated creatine kinase (CK) (e.g.,
             inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis, spinal
             muscular atrophy)

          -  Major surgery =< 3 weeks prior to starting study drugs, or persistent/ongoing side
             effects of major surgery

          -  Pregnant or nursing (lactating) women

          -  Other severe, acute, or chronic medical conditions including uncontrolled diabetes
             mellitus or psychiatric conditions or laboratory abnormalities that, in the opinion of
             the investigator, may increase the risk associated with study participation or may
             interfere with ability to participate actively in the study

          -  History of another malignancy. Participants with fully resected non-melanoma skin
             cancer, indolent early stage second malignancies, or who have been disease free for >
             three years may be eligible following discussion with study principal investigator
             (PI)

          -  Any symptomatic brain metastasis

               -  Note: Participants previously treated or untreated for this condition who are
                  asymptomatic in the absence of corticosteroid and anti-epileptic therapy are
                  allowed. Brain metastases must be stable for >= 2 weeks, with imaging (e.g.,
                  magnetic resonance imaging [MRI] or computed tomography [CT]) demonstrating no
                  current evidence of progressive brain metastases at screening)

          -  Participants receiving any medications or substances that are inhibitors or inducers
             of CYP3A enzyme(s) and CYP2C8 within 14 days of enrollment are ineligible

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions. In
             addition, these patients are at increased risk of lethal infections when treated with
             marrow-suppressive therapy