A search for a WZ resonance, in the fully leptonic final state (electrons or muons), is performed using 139 fb$^{-1}$ of data collected at a centre-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. The results are interpreted in terms of a singly charged Higgs boson of the Georgi–Machacek model, produced by WZ fusion, and of a Heavy Vector Triplet, with the resonance produced by WZ fusion or the Drell–Yan process. No significant excess over the Standard Model prediction is observed and limits are set on the production cross-section times branching ratio as a function of the resonance mass for these processes.