A search for Higgs boson pair production in events with two $b$-jets and two $\tau$-leptons is presented, using a $pp$ collision data set with an integrated luminosity of 139~\text{fb}^{-1} collected at \sqrt{s}=13~\text{TeV} by the ATLAS experiment at the LHC. Higgs boson pairs produced non-resonantly or in the decay of a narrow-width scalar resonance in the mass range 251 to 1600 GeV are targeted. Events in which at least one $\tau$-lepton decays hadronically are considered, and multivariate discriminants are used to extract the signals. No significant excess of events above the expected background is observed in the non-resonant search. The largest excess in the resonant search is observed at a resonance mass of 1 TeV, with a local (global) significance of 3.0~\sigma (2.0^{+0.4}_{-0.2}~\sigma). Observed (expected) upper limits are set at the 95% confidence-level on the non-resonant Higgs boson pair production cross-section of 4.7 (3.9) times the Standard Model prediction, assuming Standard Model kinematics, and on the resonant Higgs boson pair production cross-section between 23 and 920 fb (12 and 840 fb), depending on the heavy resonance mass.

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