  We propose models of Dark Matter that account for the 511 keV photon emission
from the Galactic Centre, compatibly with experimental constraints and
theoretical consistency, and where the relic abundance is achieved via p-wave
annihilations or, in inelastic models, via co-annihilations. Due to the Dark
Matter component that is inevitably upscattered by the Sun, these models
generically predict keV electron recoils at detectors on Earth, and could
naturally explain the excess recently reported by the XENON1T collaboration.
The very small number of free parameters make these ideas testable by detectors
like XENONnT and Panda-X, by accelerators like NA64 and LDMX, and by
cosmological surveys like the Simons observatory and CMB-S4. As a byproduct of
our study, we recast NA64 limits on invisibly decaying dark photons to other
particles.
