A search is performed for a heavy particle decaying into different-flavour, dilepton final states, using 139 fb−1 of proton-proton collision data at √s = 13 TeV collected in 2015–2018 by the ATLAS detector at the Large Hadron Collider. Final states with electrons, muons and hadronically decaying tau leptons are considered (eμ, eτ or μτ). No significant excess over the Standard Model predictions is observed. Upper limits on the production cross-section are set as a function of the mass of a Z′ boson, a supersymmetric τ-sneutrino, and a quantum black-hole. The observed 95% CL lower mass limits obtained on a typical benchmark model Z′ boson are 5.0 TeV (eμ), 4.0 TeV (eτ), and 3.9 TeV (μτ), respectively.