Let $B$ is a ball in $\mathbb R^n$, let $f$ is a diffeomorphism, mapping $B$ to a bounded region in $\mathbb R^n$. Is there a diffeomorphism $g$ of $\mathbb R^n$ such that $g=f$ when restricted to $B$?
Remark: this is a followup of the question The extension of diffeomorphism where the problem for small $B\subset Dom(f)$ has been solved. Can the problem be solved for $B=Dom(f)$?
In the question that you refer to, the problem has not been solved for small $B$, but locally in $B$. The global problem you pose cannot be solveable in general, for example in the case that $f$ is a diffeomorphism from $B$ onto $\mathbb R^n$ (which always exists, regardless of the size of $B$).