Let $R$ be a complete discrete valuation ring with finite residue field, and let $K$ be its fraction field. Let $C(R,K)$ be the (ultrametric Banach) space of continuous functions $f: R \rightarrow K$. This 1999 paper of K. Tateyama gives an analogue of Mahler's Theorem in $C(R,K)$.

Namely: let $\operatorname{Int}(R,R)$ be the ring of "integer-valued polynomials," i.e. the set of $P \in K[t]$ such that $P(R) = R$. Let $\{\psi_n(t)\}_{n=0}^{\infty}$ be a regular basis of $\operatorname{Int}(R,R)$: that is, it is a basis for $\operatorname{Int}(R,R)$ as an $R$-module and $\deg \psi_n = n$ for all $n \in \mathbb{Z}^+$.

That regular bases always exist in this context is well known (e.g. a very special case of work of Bhargava). In fact Tateyama constructs an explicit regular basis using algebraic combinations of the "Fermat quotient polynomials" $\frac{t^q-t}{\pi}$, where $q$ is the residue cardinality and $\pi$ is a uniformizer. Then he shows:

(i) For any sequence $\{a_n\}$ in $K$ [or any complete extension thereof] with $a_n \rightarrow 0$, the series $x \mapsto \sum_{n=0}^{\infty} a_n \psi_n(x)$ is uniformly convergent and thus defines an element of $C(R,K)$.

(ii) [Thm. 3.3] For any $f \in C(R,K)$, there is a unique sequence $\{a_n\}_{n=0}^{\infty}$ in $K$ such that $a_n \rightarrow 0$ and $\sum_{n=0}^{\infty} a_n \psi_n(x)$ converges uniformly to $f$.

At the end of his paper, Tateyama mentions a connection with Lubin-Tate formal groups and explains how this generalizes an earlier construction of Carlitz-Wagner in the case $R = \mathbb{F}_q[[t]]$. It seems to me at the moment that Tateyama's paper contains much of the more elementary material of de Shalit's recent paper (which leans more heavily on Lubin-Tate formal groups) referred to in Laurent Berger's answer. (Please correct me if I'm wrong; I am interested in this material of late but do not claim to have fully absorbed it.) de Shalit then goes on to discuss deeper material related to the Schneider-Teitelbaum $p$-adic Fourier theory. It is a very nice paper.

See also Chapter 3 of the 1997 book Integer-Valued Polynomials of Cahen and Chabert for a more systematic treatment of these sort of expansions from the perspective of Stone-Weierstrass theory.

Let $R$ be a complete discrete valuation ring with finite residue field, and let $K$ be its fraction field. Let $C(R,K)$ be the (ultrametric Banach) space of continuous functions $f: R \rightarrow K$. This 1999 paper of K. Tateyama gives an analogue of Mahler's Theorem in $C(R,K)$.

Namely: let $\operatorname{Int}(R,R)$ be the ring of "integer-valued polynomials," i.e. the set of $P \in K[t]$ such that $P(R) \subseteq R$. Let $\{\psi_n(t)\}_{n=0}^{\infty}$ be a regular basis of $\operatorname{Int}(R,R)$: that is, it is a basis for $\operatorname{Int}(R,R)$ as an $R$-module and $\deg \psi_n = n$ for all $n \in \mathbb{Z}^+$.

That regular bases always exist in this context is well known (e.g. a very special case of work of Bhargava). In fact Tateyama constructs an explicit regular basis using algebraic combinations of the "Fermat quotient polynomials" $\frac{t^q-t}{\pi}$, where $q$ is the residue cardinality and $\pi$ is a uniformizer. Then he shows:

(i) For any sequence $\{a_n\}$ in $K$ [or any complete extension thereof] with $a_n \rightarrow 0$, the series $x \mapsto \sum_{n=0}^{\infty} a_n \psi_n(x)$ is uniformly convergent and thus defines an element of $C(R,K)$.

(ii) [Thm. 3.3] For any $f \in C(R,K)$, there is a unique sequence $\{a_n\}_{n=0}^{\infty}$ in $K$ such that $a_n \rightarrow 0$ and $\sum_{n=0}^{\infty} a_n \psi_n(x)$ converges uniformly to $f$.

At the end of his paper, Tateyama mentions a connection with Lubin-Tate formal groups and explains how this generalizes an earlier construction of Carlitz-Wagner in the case $R = \mathbb{F}_q[[t]]$. It seems to me at the moment that Tateyama's paper contains much of the more elementary material of de Shalit's recent paper (which leans more heavily on Lubin-Tate formal groups) referred to in Laurent Berger's answer. (Please correct me if I'm wrong; I am interested in this material of late but do not claim to have fully absorbed it.) de Shalit then goes on to discuss deeper material related to the Schneider-Teitelbaum $p$-adic Fourier theory. It is a very nice paper.

See also Chapter 3 of the 1997 book Integer-Valued Polynomials of Cahen and Chabert for a more systematic treatment of these sort of expansions from the perspective of Stone-Weierstrass theory.