  A weakly coupled and light dark photon coupling to lepton charges
$L_\mu-L_\tau$ is an intriguing dark matter candidate whose coherent
oscillations alter the dispersion relations of leptons. We study how this
effect modifies the dynamics of neutrino flavor conversions, focusing on long
baseline and solar oscillations. We analyze data from the T2K, SNO, and
Super-Kamiokande experiments in order to obtain world-leading limits on the
dark photon gauge coupling for masses below $\sim 10^{-11}\,\mathrm{eV}$.
Degeneracies between shifts in the neutrino mass-squared differences and mixing
angles and the new physics effect significantly relax the current constrains on
the neutrino vacuum oscillation parameters.
