A search for resonant and non-resonant pair production of Higgs bosons in the
$b\bar{b}\tau^+\tau^-$ final state is presented. The search uses 36.1 fb$^{-1}$ of $pp$ collision data with $\sqrt{s}$ = 13 TeV recorded by the ATLAS experiment at the LHC in 2015 and 2016. Decays of the $\tau$-lepton pairs with at least one $\tau$-lepton decaying to final states with hadrons and a neutrino are considered. No significant excess above the expected background is observed in the data. The cross-section times branching ratio for non-resonant Higgs boson pair production is constrained to be less than 30.9 fb, 12.7 times the Standard Model expectation, at 95% confidence level. The data are also analyzed to probe resonant Higgs boson pair production, constraining a model with an extended Higgs sector based on two doublets and a Randall--Sundrum bulk graviton model. Upper limits are placed on the resonant Higgs boson pair production cross-section times branching ratio, excluding resonances $X$ in the mass range 305 GeV < $m_X$ < 402 GeV in the simplified hMSSM minimal supersymmetric model for $\tan\beta=2$ and excluding bulk Randall--Sundrum gravitons $G_{KK}$ in the mass range 325 GeV < $m_{G_{KK}}$ < 885 GeV for $k/\overline{M}_{Pl} = 1$.