Let $M,N$ be smooth manifolds with boundary. Let $A \subseteq N$ be closed, and let $f:A \to M$ be a smooth map.
Suppose $f$ has a continuous extension to $N$. Does it have a smooth extension to $N$?
In case the target manifold $M$ has no boundary, this is known to be true (See corollary 6.27 in Lee's book introduction to smooth manifolds).
I am asking about the case where $M$ has non-empty boundary.