So let $n,m$ be two strictly positive numbers. Let $A\subseteq\mathbf{R}^n$ be a compact $C^{\infty}$-submanifold (possibly with boundary and of real dimension not necessarily equal to $n$). Let $f:A\rightarrow \mathbf{R}^m$ be a smooth function. Then is it always possible to extend $f$ in a smooth way to all of $\mathbf{R}^n$?

Is there a good textbooks where such extension results are discussed extensively?

always. Most intro manifold theory texts have this result, but Hirsch's "Differential Topology" text goes quite far in this direction. Using the tubular neighbourhood theorem and a bump function there's rather explicit such functions you can write down. – Ryan Budney Jan 21 '11 at 4:31