Artin's approximation theorem states: "if a system of locally analytic equations in several complex variables has a formal solution then it has a locally analytic solution".
(Artin 1968, "On the solutions of analytic equations")
This is probably true for any algebraically closed field of zero characteristic with valuation? Any reference?
upd. Sorry for confusion, it seems the original Artin's theorem is already for any normed field of characteristic zero. Anyway, thanks for the comment.