On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Let $X$ and $ Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A\rightarrow Z$ . We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \rightarrow Z$ such that $F_{\mid _A}=f$ , when either (i) $X$ and $Z$ are Hilbert spaces and $X$ is separable, or (ii) $X^*$ is separable and $Z$ is an absolute Lipschitz retract, or (iii) $X=L_2$ and $Z=L_p$ with $1<p<2$ , or (iv) $X=L_p$ and $Z=L_2$ with $2<p<\infty $ , where $L_p$ is any separable Banach space $L_p(S,\Sigma ,\mu )$ with $(S,\Sigma ,\mu )$ a $\sigma $ -finite measure space.