  Extensions of the Standard Model with masses at or below the GeV scale are
motivated by searches for dark matter and precision measurements in the quark
and lepton flavour sectors, including that of the muon anomalous magnetic
moment. An excellent experimental environment to test such light new physics is
given by the Belle II experiment, which foresees to take up to 50 ab$^{-1}$ of
data. Here we consider a model with an additional gauged $U(1)_{L_\mu -
L_\tau}$ symmetry that introduces a neutral gauge boson, a Dark Photon, with
possibly large couplings to muon- and tau-flavored leptons, including
neutrinos. Dark Photon mixing with the Standard Model photon is loop induced,
allowing it to couple to electrically charged fermions other than muons and
taus. We systematically investigate the possible search strategies for Dark
Photons with four fermions final states. We identified search channels with
muons as the most promising ones, and we analyse the kinematic distributions to
obtain cuts that optimise the sensitivity of Belle II searches for the Dark
Photon. Summarising the sensitivities from the most promising search channels
we provide a comprehensive overview of future searches at Belle II.
