We present a dark sector model based on gauging the $L_\mu - L_\tau$ symmetry
that addresses anomalies in $b \rightarrow s \mu^+ \mu^-$ decays and that
features a particle dark matter candidate. The dark matter particle candidate
is a vector-like Dirac fermion coupled to the $Z^\prime$ gauge boson of the
$L_{\mu}-L_{\tau}$ symmetry. We compute the dark matter thermal relic density,
its pair-annihilation cross section, and the loop-suppressed dark
matter-nucleon scattering cross section, and compare our predictions with
current and future experimental results. We demonstrate that after taking into
account bounds from $B_s$ meson oscillations, dark matter direct detection, and
the CMB, the model is highly predictive: $B$ physics anomalies and a viable
particle dark matter candidate, with a mass of $\sim (5-23)$~GeV, can be
accommodated only in a tightly-constrained region of parameter space, with
sharp predictions for future experimental tests. The viable region of parameter
space expands if the dark matter is allowed to have $L_\mu-L_\tau$ charges that
are smaller than those of the SM leptons.
