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.