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So far, the observed effects of dark matter are compatible with it having purely gravitational interactions. However, in many models, dark matter has additional interactions with itself, with the Standard Model, and/or with additional hidden sector states. In this paper, we discuss models in which dark matter interacts through a light vector mediator, giving rise to a long-ranged force between dark matter particles. Coherent scattering effects through this force can lead to the exponential growth of small perturbations, in analogy to electromagnetic plasma instabilities. These instabilities can be significant at couplings many orders of magnitude below those for which the usual particle-by-particle constraints on dark matter self-interactions apply. While this possibility has been noted in the literature, we provide the first systematic study of such instabilities, including the case where the mediator has finite mass. The latter is relevant for models of kinetically mixed `dark photon' mediators, which represent an important target for proposed dark matter detection experiments. Our analyses are of the growth of small perturbations, so do not immediately provide observational constraints on dark matter models - however, they do motivate further study of large regions of parameter space.