Interspecies hybridization has been recently recognized as potentially common in wild animals, but the extent to which it shapes modern genomes is still poorly understood. Distinguishing historical hybridization events from other processes leading to phylogenetic discordance among different markers requires a well-resolved species tree that considers all modes of inheritance, and overcomes systematic problems due to rapid lineage diversification by sampling large genomic character sets. Here we assessed genome-wide phylogenetic variation across a diverse mammalian family, Felidae (cats). We combined genotypes from a genome-wide SNP array with additional autosomal, X- and Y-linked variants to sample ~150 kilobases of nuclear sequence, in addition to complete mitochondrial genomes generated using light-coverage Illumina sequencing. We present the first robust felid timetree that accounts for unique maternal, paternal, and biparental evolutionary histories. Signatures of phylogenetic discordance were abundant in the genomes of modern cats, in many cases indicating hybridization as the most likely cause. Comparison of big cat whole-genome sequences revealed a substantial reduction of X-linked divergence times across several large recombination coldspots, which were highly enriched for signatures of selection-driven post-divergence hybridization between the ancestors of the snow leopard and lion lineages. These results highlight the mosaic origin of modern felid genomes and the influence of sex chromosomes and sex-biased dispersal in post-speciation gene flow. A complete resolution of the Tree of Life will require comprehensive genomic sampling of biparental and sex-limited genetic variation to identify and control for phylogenetic conflict caused by ancient admixture and sex-biased differences in genomic transmission.