A search is performed for a heavy particle decaying into different flavour dilepton pairs (eμ, eτ or μτ), using 3.2 fb−1 of proton–proton collision data at √s=13 TeV collected in 2015 by the ATLAS detector at the Large Hadron Collider. No excess over the Standard Model prediction is observed. Limits at the 95 % credibility level are set on the mass of a Z′ boson with lepton-flavour-violating couplings at 3.0, 2.7 and 2.6 TeV, and on the mass of a supersymmetric τ sneutrino with R-parity-violating couplings at 2.3, 2.2 and 1.9 TeV, for eμ, eτ and μτ final states, respectively. The results are also interpreted as limits on the threshold mass for quantum black hole production.