A search for a heavy resonance decaying into $WZ$ in the fully leptonic channel (electrons and muons) is performed. It is based on proton–proton collision data collected by the ATLAS experiment at the Large Hadron Collider at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 36.1 fb$^{-1}$. Limits are set on the production cross section times branching ratio of a heavy vector particle produced either in quark–antiquark fusion or through vector-boson fusion. Constraints are also obtained on the mass and couplings of a singly charged Higgs boson, in the Georgi–Machacek model, produced through vector-boson fusion.