Let $AE(X)$ denote the Arens-Eells space on a Banach space $X$.  Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow AE(X)
\\
x&\mapsto \delta_x
\end{aligned}
$$
Is the map $\delta$ ever Gateau (or Fréchet) differentiable?


---
Recall that $AE(X)$ is the/a [pre-dual][1] of the space $Lip_0(X)$ whose elements are Lipschitz functions sending $0\in X$ to $0\in \mathbb{R}$, $\delta_x$ denotes the evaluation map defined on Lipschitz functions $f\in Lip_0(X)$ by
$$
\delta_x(f):= f(x),
$$
and $AE(X)$ is normed by:
$$
\|F-G\|:=\inf_{f \in Lip_0(X),\, Lip(f)\leq 1} F(f)-G(f),
$$
where $Lip(f)$ denotes the Lipschitz constant.


  [1]: https://arxiv.org/abs/1611.01812