Let $G$ be a topological group, and $(X, d)$ a metric space. The function $f : G \to X$ is left uniformly continuous if for all $\varepsilon > 0$ there exists an identity neighbourhood $U$ such that for all $g\in G, u \in U$ it holds $d(f(g), f(ug)) < \varepsilon$.
If $G$ is compact or discrete, every continuous function is uniformly continuous. Is there a larger class of groups for which this holds, maybe with some extra assumptions on $X$ or $f$? I am particularly interested in totally disconnected locally compact groups $G$, ultrametric normed vector spaces $X$ and bounded functions $f$.