I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following
Question. Let $X$ and $Y$ be metric spaces. When each continuous function from $X$ to $Y$ can be approximated by Lipschitz functions in the topology of uniform convergence on compact sets?
It seems that I can prove that the question has a positive answer provided a pair $(X,Y)$ satisfies the following
Condition. Each bounded (that is with finite diameter of the image) function from $A$ to $Y$ defined on a uniformly discrete subset $A$ of the space $X$ can be extended to a Lipschitz function from $X$ to $Y$. (I call a subset $A$ of a metric space $(X,d)$ uniformly discrete, provided there exists a number $\varepsilon>0$ such that $d(x,y)\ge\varepsilon$ for each distinct points $x,y\in A$.) In particular, Valentine Theorem (see, [p. 11, LR] or [V]) implies that a pair $(X,Y)$ satisfies Condition provided $X$ and $Y$ are Hilbert spaces.
We found no references on the above results and I think about a publication and a continuation of this investigation. But I am not a specialist in this domain.
So, I care about the following questions. Are the above results new, good and worthy to be published somewhere? What another related problems are worthy to be investigated? In particular, is it known which pairs $(X,Y)$ of metric spaces satisfies Condition?
[LR] Ronnie Levy, Michael David Rice. The approximation and extension of uniformly continuous Banach space valued mappings, Commentationes Mathematicae Universitatis Carolinae, 24:2 (1983), 251--265.
[V] F.A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math., 67 (1945), 83--93.