  In this paper, we study the Vlasov-Maxwell system in the non-relativistic
limit, that is in the regime where the speed of light is a very large
parameter. We consider data lying in the vicinity of homogeneous equilibria
that are stable in the sense of Penrose (for the Vlasov-Poisson system), and
prove Sobolev stability estimates that are valid for times which are polynomial
in terms of the speed of light and of the inverse of size of initial
perturbations. We build a kind of higher-order Vlasov-Darwin approximation
which allows us to reach arbitrarily large powers of the speed of light.
