Let $X,Y$ be smooth manifolds.
Let $F \subset G$ denote the spaces of, respectively, smooth and continuous functions $X\rightarrow Y$; we give $F,G$ the compact-open topologies. (This should be compatible with the subspace topology on $F$ from $G$.)
Since every continuous map is homotopic to a smooth one by e.g. Whitney approxiomation, we know that $F$ meets every path-component of $G$.
I was wondering, is $F$ furthermore a deformation-retract of $G$?