Yes, even more is true. The argument is as follows. Let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact set. Then $f|_K$ is uniformly continuous, let $\omega$ be its nondecreasing subadditive modulus of continuity. By [McShane-Whitney's extension formula][1] $f|_K$ admits an uniformly continuous extension to $X$ with the same modulus, more concretely $$F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$$ is this extension which is uniformly continuous on the whole $X$.


  [1]: https://en.wikipedia.org/wiki/Whitney_extension_theorem