Here is Ryan Budney's answer from the comments, I'm copying it here so that this question does not re-appear on the front page as unanswered.

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Let $f:\mathbb{R}\to\mathbb{R}$ be the absolute value function. It is piecewise smooth for some triangulation of $\mathbb{R}$, and smooth on any subcomplex not containing the origin. But any smooth approximation to $f$ can not be close to $f$ in the Lipschitz norm. You could construct a version of this for compact manifolds, replace $\mathbb{R}$ with $[−1,1]$ for example.