  A search is performed for a heavy particle decaying into different-flavour,
dilepton final states, using 139 $\mathrm{fb}^{-1}$ of proton-proton collision
data at $\sqrt{s}=13$ 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 ($e\mu$, $e\tau$ or $\mu\tau$). 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 $\tau$-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 (e$\mu$), 4.0 TeV (e$\tau$), and 3.9 TeV ($\mu\tau$),
respectively.
