Characterization of p-adic analytic functions

Question

Is there a characterization of which p-adic functions, $f(x)$, can be expressed as a power series in $x$? I'm thinking something akin to "all infinitely differentiable functions can be expressed as a convergent power series" (I’m not sure if this is true).

Answer 1

You're clearly interested in an answer more substantial than "a function is analytic if and only if it is analytic".

That infinite differentiability should imply (local) analyticity is false over $\mathbf R$, so it's not surprising this shouldn't work over $\mathbf Q_p$ as well. That it works over $\mathbf C$ is kind of a miracle.

Let's take a cue from classical Fourier series on $\mathbf R$, where the rate of decay of the Fourier coefficients is related to the amount of smoothness of the function (I'm not saying there is an equivalence, but there's definitely a relationship). We look around for series expansions of $p$-adic functions, and the Mahler expansion for continuous functions turns out to be nice in this respect. The continuous functions $f : \mathbf Z_p \to \mathbf Q_p$ are precisely those that admit a series representation $f(x) = \sum a_n\binom{x}{n}$ for all $x$ in $\mathbf Z_p$ where $|a_n|_p \to 0$ as $n \to \infty$. That series is the Mahler expansion of $f$. It turns out that $f$ is $p$-adic analytic, meaning it is expressible as a single power series $\sum_{n \geq 0} c_nx^n$ on all of $\mathbf Z_p$, if and only if its Mahler coefficients $a_n$ satisfy the relation $|a_n/n!|_p \to 0$ as $n \to \infty$. For a proof, see Theorem 4.4 here.