$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$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 Gâteaux (or Fréchet) differentiable?


---
Recall that $\AE(X)$ is the/a [pre-dual][1] of the Banach space $\Lip_0(X)$ whose elements are Lipschitz functions sending $0\in X$ to $0\in \mathbb{R}$ (with norm sending any $f\in \Lip_0(X)$ to its (unique) [Lipschitz constant][2] $\Lip(f)$), $\delta_x$ denotes the evaluation map defined on Lipschitz functions $f\in \Lip_0(X)$ by
$$
\delta_x(f):= f(x),
$$
and $\AE(X)$ is normed using the [dual-norm][3] construction; i.e.:
$$
\|F-G\|:=\inf_{f \in \Lip_0(X),\, \Lip(f)\leq 1} F(f)-G(f).
$$



  [1]: https://arxiv.org/abs/1611.01812
  [2]: https://en.wikipedia.org/wiki/Lipschitz_continuity
  [3]: https://en.wikipedia.org/wiki/Dual_norm