2 Added two remarks expanding on those of Kevin Buzzard and dke

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.

1

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.