Can annihilating dark matter be lighter than a few GeVs?

We estimate the gamma-ray fluxes from the residual annihilations of dark matter (DM) particles having a mass m_dm [MeV, O(10) GeV] (a possible solution to the DM issue provided they have a new kind of interactions and no significant coupling to the Z) and compare them to observations. We find that particles lighter than O(100 MeV) can be viable DM candidates provided their dominant annihilation cross section is S-wave suppressed so as to satisfy the gamma-ray constraints. A similar conclusion is obtained for particles lighter than O(10) GeV from the study of radio fluxes, assuming a NFW profile and that they mainly annihilate into electrons.