Charged Higgs bosons produced in association with a single top quark and decaying via H±→τν are searched for with the ATLAS experiment at the LHC, using proton–proton collision data at s=13 TeV corresponding to an integrated luminosity of 3.2 fb−1. The final state is characterised by the presence of a hadronic τ decay and missing transverse momentum, as well as a hadronically decaying top quark, resulting in the absence of high-transverse-momentum electrons and muons. The data are found to be consistent with the expected background from Standard Model processes. A statistical analysis leads to 95% confidence-level upper limits on the production cross section times branching fraction, σ(pp→[b]tH±)×BR(H±→τν), between 1.9 pb and 15 fb, for charged Higgs boson masses ranging from 200 to 2000 GeV. The exclusion limits for this search surpass those obtained with the proton–proton collision data recorded at s=8 TeV.