We study $b \to s \ell_1^+ \ell_2^-$ transitions, both for the lepton flavour conserving $\ell_1=\ell_2$ and violating case $\ell_1 \neq \ell_2$, in a minimal extension of the Standard Model proposed in \cite{Aebischer:2019blw}. In this framework, the Standard Model (SM) gauge group is enlarged by a new $U(1)^\prime$ component. The fermion $U(1)^\prime$ charges are assigned in a generation-dependent way, and involve three rational parameters $\epsilon_{1,2,3}$ summing to zero by the condition of cancellation of the gauge anomalies. Each $\epsilon_i$ is common to all fermions in a generation, which produces correlations among quark and lepton observables. The new neutral gauge boson $Z^\prime$ has flavour violating couplings to quarks and leptons. For SM allowed processes, small deviations with respect to the SM predictions are found: this is a consequence of a feature of the model where quark and lepton sectors preclude each other large deviations from SM. Lepton flavour violating processes are allowed at tree-level. The experimental upper bounds for the rates of the processes $\tau^- \to \mu^- \mu^+ \mu^-$, $\mu^- \to e^- \gamma$, $\mu^- \to e^- e^+ e^-$ and the $ \mu^- \to e^-$ conversion in nuclei play a hierarchical role in constraining the branching fractions of lepton flavour violating $B$ and $B_s$ decays.