Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets

Let $E$, $F$ be separable Hilbert spaces, and assume that $E$ is infinite-dimensional. We show that for every continuous mapping $f:E\to F$ and every continuous function $\varepsilon: E\to (0, \infty)$ there exists a $C^{\infty}$ mapping $g:E\to F$ such that $\|f(x)-g(x)\|\leq\varepsilon(x)$ and $Dg(x):E\to F$ is a surjective linear operator for every $x\in E$. We also provide a version of this result where $E$ can be replaced with a Banach space from a large class (including all the classical spaces with smooth norms, such as $c_0$, $\ell_p$ or $L^{p}$, $1<p<\infty$), and $F$ can be taken to be any Banach space such that there exists a bounded linear operator from $E$ onto $F$. In particular, for such $E, F$, every continuous mapping $f:E\to F$ can be uniformly approximated by smooth open mappings. Part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from Banach spaces or Hilbert manifolds.