Let $f: \mathbb Z_{\ge 0} \to \mathbb C_p$ be any function. My understanding is that Mahler's theorem says that $f$ extends to a continuous function $f: \mathbb Z_p \to \mathbb C_p$ if and only if the Mahler coefficients \begin{align*} d_0 &= f(0) \\ d_1 &= f(1)-f(0) \\ d_2 &= f(2)-2f(1)+f(0) \\ &\vdots \\ d_n &= \sum_k (-1)^{n-k} \binom nk f(k) \end{align*} tend to zero in the $p$-adic sense. By finite differences, this is equivalent to the existence of a polynomial $P_e \in \mathbb Q[x]$, taking only integer values on $\mathbb Z$, for which $f(n) \equiv P_e(n) \pmod{p^e}$ holds for $n \ge 0$. (Please correct me if this is wrong!) In either case, we have the explicit formula $$f(x) = \sum_k d_k \binom xk \qquad x \in \mathbb Z_p.$$

In reading proofs of Skolem-Mahler-Lech theorem (e.g. here), I have the following additional questions.

Are there conditions (in terms of either $d_n$ or $P_e$) on when the extended function $f : \mathbb Z_p \to \mathbb C_p$ is

*analytic*?Assuming $f$ isn't identically zero, are there conditions on which $f$ can have only finitely many zeros? (In particular, the blog post linked seems to claim that analytic is sufficient, but as $\mathbb Z_p$ is quite disconnected I'm confused why this is true.)