  A search for high-mass resonances decaying into a $\tau$-lepton and a
neutrino using proton-proton collisions at a center-of-mass energy of
$\sqrt{s}=13$ TeV is presented. The full Run 2 data sample corresponding to an
integrated luminosity of 139 fb$^{-1}$ recorded by the ATLAS experiment in the
years 2015-2018 is analyzed. The $\tau$-lepton is reconstructed in its hadronic
decay modes and the total transverse momentum carried out by neutrinos is
inferred from the reconstructed missing transverse momentum. The search for new
physics is performed on the transverse mass between the $\tau$-lepton and the
missing transverse momentum. No excess of events above the Standard Model
expectation is observed and upper exclusion limits are set on the $W^\prime\to
\tau \nu$ production cross-section. Heavy $W^\prime$ vector bosons with masses
up to 5.0 TeV are excluded at 95% confidence level, assuming that they have the
same couplings as the Standard Model $W$ boson. For non-universal couplings,
$W^\prime$ bosons are excluded for masses less than 3.5-5.0 TeV, depending on
the model parameters. In addition, model-independent limits on the visible
cross-section times branching ratio are determined as a function of the lower
threshold on the transverse mass of the $\tau$-lepton and missing transverse
momentum.
