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 (\(\Delta a_{\mu,e}\)) 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 \(\Delta a_{\mu}\), albeit subject to stringent experimental constraints. Here we explore a minimal extension of the Standard Model via a leptophilic vector boson \(Z^\prime\), 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 \(\tau\) decays) and from rare charged lepton flavour violating transitions. Once these are accommodated, one can saturate the tensions in \(\Delta a_{\mu}\), but \(\Delta a_{e}\) 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 \(Z^\prime\) at a future muon collider, emphasising the role of the \(\mu^+\mu^- \to\tau^+\tau^- \) forward-backward asymmetry as a key probe of the model.