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 \(1Banach space \(L_p(S,\Sigma ,\mu )\) with \((S,\Sigma ,\mu )\) a \(\sigma \)-finite measure space.