When does a p-adic function have a Mahler expansion?

It is false for the valuation ring in any nontrivial finite extension of $\mathbb{Q}_p$. The coefficients of the Mahler expansion of a continuous function $\mathcal{O} \to \mathbb{C}_p$ are determined by its restriction to $\mathbb{Z}_p$ (they are given as $n$-th differences of the sequence of values on nonnegative integers, in fact). But there are different continuous functions $\mathcal{O} \to \mathbb{C}_p$ with the same restriction to $\mathbb{Z}_p$.

Even worse, the Mahler expansions need not even converge because if $x$ is not in $\mathbb{Z}_p$, the binomial coefficient values may have negative valuation.

EDIT: As Kevin Buzzard and dke suggest, one can give a positive answer if your question is interpreted differently. The point of this edit is to make a few explicit remarks in these two directions.

1) If it is known in advance that $f \colon \mathcal{O} \to \mathbb{C}_p$ is represented by a single convergent power series, then the Mahler expansion of $f|_{\mathbb{Z}_p}$ converges to $f$ on all of $\mathcal{O}$. This can be deduced from the theorem that a continuous function $\mathbb{Z}_p \to \mathbb{C}_p$ is analytic if and only if the Mahler expansion coefficients $a_n$ satisfy $a_n/n! \to 0$ (see Theorem 54.4 in Ultrametric calculus: an introduction to $p$-adic analysis by W. H. Schikhof).

2) If one chooses a $\mathbb{Z}_p$-basis of $\mathcal{O}$, then $f$ can be interpreted as a continuous function $\mathbb{Z}_p^r \to \mathbb{C}_p$, and any such function has a multivariable Mahler expansion $$\sum a_n \binom{x_1}{n_1} \cdots \binom{x_r}{n_r},$$ where the sum is over tuples $n=(n_1,\ldots,n_r)$ with $n_i \in \mathbb{Z}_{\ge 0}$, and $a_n \to 0$ $p$-adically.

Ehud de Shalit has a preprint called "Mahler's theorem for local fields" which does what you want.

As Bjorn says in his answer, the set of binomial coefficient functions just isn't sufficient in general. However, plenty has been written about analogues of Mahler expansions, i.e. finding nice bases for various spaces of continuous functions, going back to Amice in the 1960's for finite extensions of ${\mathbb Q}_p$, as well as positive characteristic versions. This is all very nicely explained in Keith Conrad's The Digit Principle, J. Number Theory 84 (2000), no. 2, 230--257. arXiv version

S. Evrard has recently extended some of these results to cases with infinite residue field Normal bases of rings of continuous functions constructed with the $(q_n)$-digit principle. Acta Arith. 135 (2008), no. 3, 219--230.

Edited to add: probably of more relevance to your question though would be the theory of Mahler-type expansions developed in p-Adic Fourier Theory by Schneider and Teitelbaum.