Suppose $f: \partial M \to S^n$ is a smooth map, is it always possible to extend it to a map on $M$?
If $dim M > n$ it is not always possible for sure. Just consider $\overline{B}^m$. What about $dim M \leq n$? The usual partition of unity argument does not work because one cannot guarantee that the extension stays in $S^n$.
If $\dim M\leq n,$ then $\dim\partial M<n=\dim S^n,$ and this implies that $f$ is not surjective. So, you can think of $f$ as a map to $\mathbb{R}^n$, and as such, it can be extended from $\partial M$ to $M$.