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.