  We prove an implicit function theorem for C^k-maps from arbitrary topological
vector spaces over valued fields to Banach spaces (for k at least 2). As a
tool, we show the C^k-dependence of fixed points on parameters for suitable
families of contractions of a Banach space. Similar results are obtained for k
times strictly differentiable maps, and for k times Lipschitz differentiable
maps. In the real case, our results subsume an implicit function theorem for
Keller C^k_c-maps from arbitrary topological vector spaces to Banach spaces.
