In Bourbaki "Variétés différentielles et analytiques" there is a statement (without proof) that for a vector field on smooth manifold over complete normed field (characteristic zero) there is a local unique flow along it. Is it true or not? Is there any proof for this statement?
In the p-adic case, the standard theory deals not with smooth functions and manifolds but with analytic ones. A complete exposition including necessary results from analysis is given by J.-P. Serre, "Lie algebras and Lie groups", New York, Benjamin, 1965.