  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 $B$ into a $\tau$-lepton and $b$-quark. For scalar
leptoquarks, masses below 1460 GeV are excluded assuming $B=100$%, while for
vector leptoquarks the corresponding limit is 1650 GeV (1910 GeV) in the
minimal-coupling (Yang-Mills) scenario.
