I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one.
Let $M$ be a smooth $n$-manifold with boundary $N:= \partial M$, and assume that there exists a continuous retraction of $M$ onto $N$, namely, a continuous map $r \colon M \to N$ that is the identity on $N$.
Question. Does there exist a smooth retraction $s \colon M \to N$? And what about a smooth retraction homotopic to $r$?
I am aware that, by Whitney Approximation Theorem, $r \colon M \to N$ is homotopic to a smooth map, but I do not see how to conclude from this that it is homotopic to a smooth retraction.
Motivation. This question arose when I was trying to mimic the standard proof of Brouwer Fixed Point Theorem (for continuous maps $f \colon \mathbb{D}^n \to \mathbb{D}^n$) by using de Rham cohomology, instead of singular homology. In the homology setting, the starting point is the non-existence of a continuous retraction $r \colon \mathbb{D}^n \to S^{n-1}$, which is obtained by contradiction looking at the functorial group homomorphisms induced by $r$ and by the inclusion $i \colon S^{n-1} \to \mathbb{D}^n$. But de Rham cohomology is only functorial with respect to smooth maps...