  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.
