SF3B1 is the most commonly mutated RNA splicing factor in cancer^{, , –}, but the mechanisms by which SF3B1 mutations promote malignancy are poorly understood. Here we integrated pan-cancer splicing analyses with a positive-enrichment CRISPR screen to prioritize splicing alterations that promote tumorigenesis. We report that diverse SF3B1 mutations converge on repression of BRD9, which is a core component of the recently described non-canonical BAF chromatin-remodelling complex that also contains GLTSCR1 and GLTSCR1L^{, –}. Mutant SF3B1 recognizes an aberrant, deep intronic branchpoint within BRD9 and thereby induces the inclusion of a poison exon that is derived from an endogenous retroviral element and subsequent degradation of BRD9 mRNA. Depletion of BRD9 causes the loss of non-canonical BAF at CTCF-associated loci and promotes melanomagenesis. BRD9 is a potent tumour suppressor in uveal melanoma, such that correcting mis-splicing of BRD9 in SF3B1-mutant cells using antisense oligonucleotides or CRISPR-directed mutagenesis suppresses tumour growth. Our results implicate the disruption of non-canonical BAF in the diverse cancer types that carry SF3B1 mutations and suggest a mechanism-based therapeutic approach for treating these malignancies.