A search for pair-produced scalar or vector leptoquarks decaying into a b-quark and a $\tau $-lepton is presented using the full LHC Run 2 (2015–2018) data sample of 139 fb$^{-1}$ collected with the ATLAS detector in proton–proton collisions at a centre-of-mass energy of $\sqrt{s} =13$ TeV. Events in which at least one $\tau $-lepton decays hadronically are considered, and multivariate discriminants are used to extract the signals. No significant deviations from the Standard Model expectation are observed and 95% confidence-level upper limits on the production cross-section are derived as a function of leptoquark mass and branching ratio $\mathcal {B}$ into a $\tau $-lepton and b-quark. For scalar leptoquarks, masses below 1460 GeV are excluded assuming $\mathcal {B}=100$%, while for vector leptoquarks the corresponding limit is 1650 GeV (1910 GeV) in the minimal-coupling (Yang–Mills) scenario.