The apparent tensions emerging from the comparison of experimental data of the anomalous magnetic moments of the muon and electron to the Standard Model predictions ( $$\Delta a_{\mu ,e}$$ $\Delta a_{\mu ,e}$ ) could be interpreted as a potential signal of New Physics. Models encompassing a light vector boson have been known to offer a satisfactory explanation to $$\Delta a_{\mu }$$ $\Delta a_{\mu }$ , albeit subject to stringent experimental constraints. Here we explore a minimal extension of the Standard Model via a leptophilic vector boson $$Z^\prime $$ $Z^{\prime }$ , under the hypothesis of strictly flavour-violating couplings of the latter to leptons. The most constraining observables to this ad-hoc construction emerge from lepton flavour universality violation (in Z and $$\tau $$ $\tau$ decays) and from rare charged lepton flavour violating transitions. Once these are accommodated, one can saturate the tensions in $$\Delta a_{\mu }$$ $\Delta a_{\mu }$ , but $$\Delta a_{e}$$ $\Delta a_e$ is predicted to be Standard Model-like. We infer prospects for several observables, including leptonic Z decays and several charged lepton flavour violating processes. We also discuss potential signatures of the considered $$Z^\prime $$ $Z^{\prime }$ at a future muon collider, emphasising the role of the $$\mu ^+\mu ^- \rightarrow \tau ^+\tau ^- $$ ${\mu }^{+}{\mu }^{-}\to {\tau }^{+}{\tau }^{-}$ forward-backward asymmetry as a key probe of the model.