We present a model to explain LHCb's recent measurements of $RK$ and $RK\ast$ based on an anomaly-free, spontaneously-broken $U(1)F$ gauge symmetry, without any fermionic fields beyond those of the Standard Model (SM). The model explains the hierarchical heaviness of the third family and the smallness of quark mixing. The $U(1)F$ charges of the third family of SM fields and the Higgs doublet are set equal to their respective hypercharges. A heavy $Z\prime$ particle with flavour-dependent couplings can modify the $[bL―\gamma \rho sL][\mu L―\gamma \rho \mu L]$ effective vertex in the desired way. The $Z\prime$ contribution to $Bs-Bs―$ mixing is suppressed by a small mixing angle connected to $Vts$, making the constraint coming from its measurement easier to satisfy. The model can explain $RK$ and $RK(\ast )$ whilst simultaneously passing other constraints, including measurements of the lepton flavour universality of $Z$ couplings.