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 padic 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. 

