$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$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 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 $\Lip(f)$), $\delta_x$ 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 construction; i.e.: $$ \|F-G\|:=\inf_{f \in \Lip_0(X),\, \Lip(f)\leq 1} F(f)-G(f). $$