On the Tietze extension theorem

If f is a uniformly continuous function defined on a closed subset A of a metric space with values in [1,2], its extension given by the formula:

F(x)= inf ( f(a)d(x,a) : a in A) / d(x,A)

is uniformly continuous as well: this is proven in a paper of Mandelkern 'on the uniform continuity of Tietze's extensions.

I am wondering what happens if one replaces "uniformly continuous" by Lipschitzean (allowing a different Lipschitz-constant for the extension).

I am tempted to believe it's wrong, but cannot find any couterexample...

Many thanks!

PS under the hypotheses of continuity only, this is well known; in many textbooks this formula is used often to provide an explicit extension; in the case of Lipschitzean functions, there are other ways to extend it to a Lipshcitzean function with the same constant, which are more natural. So this question, in this sense, is not "natural", yet correctly settled...