We attempt to explain recent anomalies in semileptonic $B$ decays at LHCb via a composite Higgs model, in which both the Higgs and an $SU(2)L$-triplet leptoquark arise as pseudo-Goldstone bosons of the strong dynamics. Fermion masses are assumed to be generated via the mechanism of partial compositeness, which largely determines the leptoquark couplings and implies non-universal lepton interactions. The latter are needed to accommodate tensions in the $b\to s\mu \mu$ dataset and to be consistent with a discrepancy measured at LHCb in the ratio of $B+\to K+\mu +\mu -$ to $B+\to K+e+e-$ branching ratios. The data imply that the leptoquark should have a mass of around a TeV. We find that the model is not in conflict with current flavour or direct production bounds, but we identify a few observables for which the new physics contributions are close to current limits and where the leptoquark is likely to show up in future measurements. The leptoquark will be pair-produced at the LHC and decay predominantly to third-generation quarks and leptons, and LHC13 searches will provide further strong bounds.