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](https://cfranc.wordpress.com/2010/03/18/the-skolem-mahler-lech-theorem/)),
I have the following additional questions.


1. 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*?
2. 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.)



Analytic p-adic functions from Mahler coefficients
For your first question: Theorem 54.4 (page 166) in Schikhof's "Ultrametric Calculus" says that $f$ is analytic if and only if


$$\lim\_{n\to\infty}\frac{d\_n}{n!}=0.$$


Here, analytic means that $f$ has a power series expansion which converges for all $|x|\le1$ (in $\mathbb{Q}\_p$, hence in $\mathbb{C}\_p$).


Actually, in his book Schikhof gives characterizations in terms of Mahler coefficients of a lot of interesting spaces of $p$-adic functions. Many of them are due to Yvette Amice.


If your meaning of analytic is "Krasner analytic", you may look at Robert's "A course in $p$-adic analysis", section 4.3, page 344.


For your second question, as Asvin points out, a nonzero power series on the closed unit ball has only finitely many zeros there ("Strassman theorem", see Robert, page 306).


