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