Suppose we have a function $f$ defined on a disk in the field $\mathbb Q_p$, given by an absolutely convergent power series, with $f(0)=0$ and $f'(0)=1$:
$$\mathbb Q_p\ni f(x)=x+\sum_{n\geq2}a_nx^n,\quad|x|\leq R>0$$ $$\mathbb R\ni\sum_{n\geq2}|a_n|R^n<\infty$$
Are there disks of (positive) radii $S$ and $r\leq R$, and a function $g$, also given by an absolutely convergent power series, such that $g(f(x))=x$ for all $|x|\leq r$, and $f(g(y))=y$ for all $|y|\leq S$?
$$g(y)=y+\sum_{n\geq2}b_ny^n,\quad|y|\leq S$$ $$\sum_{n\geq2}|b_n|S^n<\infty$$
(Assuming it is true) I'd actually like a uniform proof that works for $\mathbb R$, $\mathbb C$, or $\mathbb Q_p$. See Is the inverse of a real analytic function still analytic? and If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse. for proofs specific to the complex field. See Composition of real-analytic functions is real-analytic for a uniform proof of the type I'm looking for.
By expanding the formal power series $g(f(X))$, I can see that the required coefficients $b_n$ exist and are unique. But I can't see that $g(y)$ converges for some $y\neq0$.