Thermal dark matter at the MeV scale faces stringent bounds from a variety of cosmological probes. Here we perform a detailed evaluation of BBN bounds on the annihilation cross section of dark matter with a mass 1 MeV lesssim m_χ lesssim 1 GeV. For p-wave suppressed annihilations, constraints from BBN turn out to be significantly stronger than the ones from CMB observations, and are competitive with the strongest bounds from other indirect searches. We furthermore update the lower bound from BBN on the mass of thermal dark matter using improved determinations of primordial abundances. While being of similar strength as the corresponding bound from CMB, it is significantly more robust to changes in the particle physics model.