Extension of a self-diffeomorphism of unit ball

Question

Let $B$ be the unit open ball in $\mathbb R^n$ and $\phi$ a self diffeomorphism of $B$. Can we find a self diffeomorphism $f$ of $\mathbb R^n$ such that $$ f=\phi $$ on $B$?

Answer 1

Certainly not, without making some further assumptions about $ f $. Consider, for example $ f(x) = \left( 1 - \sqrt{1 - |x|^2} \right) \ x $. Any extension cannot be differentiable on the boundary $ |x| = 1 $.