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

Dark matter models for the 511 keV galactic line predict keV electron recoils on Earth