  We recently proposed a method to constrain $s$-wave annihilating MeV dark
matter from a combination of the Voyager 1 and the AMS-02 data on cosmic-ray
electrons and positrons. Voyager 1 actually provides an unprecedented probe of
dark matter annihilation to cosmic rays down to $\sim 10$ MeV in an energy
range where the signal is mostly immune to uncertainties in cosmic-ray
propagation. In this article, we derive for the first time new constraints on
$p$-wave annihilation down to the MeV mass range using cosmic-ray data. To
proceed, we derive a self-consistent velocity distribution for the dark matter
across the Milky Way by means of the Eddington inversion technique and its
extension to anisotropic systems. As inputs, we consider state-of-the-art
Galactic mass models including baryons and constrained on recent kinematic
data. They allow for both a cored or a cuspy halo. We then calculate the flux
of cosmic-ray electrons and positrons induced by $p$-wave annihilating dark
matter and obtain very stringent limits in the MeV mass range, robustly
excluding cross sections greater than $\sim 10^{-22}{\rm cm^3/s}$ (including
theoretical uncertainties), about 5 orders of magnitude better than current CMB
constraints. This limit assumes that dark matter annihilation is the sole
source of cosmic rays and could therefore be made even more stringent when
reliable models of astrophysical backgrounds are included.
