A search is performed for a heavy particle decaying into different flavour dilepton pairs ([Formula: see text], [Formula: see text] or [Formula: see text]), using 3.2 fb[Formula: see text] of proton-proton collision data at [Formula: see text] 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 [Formula: see text] boson with lepton-flavour-violating couplings at 3.0, 2.7 and 2.6 TeV, and on the mass of a supersymmetric [Formula: see text] sneutrino with R-parity-violating couplings at 2.3, 2.2 and 1.9 TeV, for [Formula: see text], [Formula: see text] and [Formula: see text] final states, respectively. The results are also interpreted as limits on the threshold mass for quantum black hole production.