A generic search for resonances is performed with events containing a Z boson with transverse momentum greater than 100 GeV, decaying into e$^{+}$e$^{−}$ or μ$^{+}$μ$^{−}$. The analysed data collected with the ATLAS detector in proton-proton collisions at a centre-of-mass energy of 13 TeV at the Large Hadron Collider correspond to an integrated luminosity of 139 fb$^{−1}$. Two invariant mass distributions are examined for a localised excess relative to the expected Standard Model background in six independent event categories (and their inclusive sum) to increase the sensitivity. No significant excess is observed. Exclusion limits at 95% confidence level are derived for two cases: a model-independent interpretation of Gaussian-shaped resonances with the mass width between 3% and 10% of the resonance mass, and a specific heavy vector triplet model with the decay mode W′ → ZW →ℓℓqq.