Let $\mathcal{M}(X)$ denote the Banach space of finite *signed* Borel measures on a Banach space $X$, normed by the *total-variation norm*.  Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow M(X)
\\
x&\mapsto \delta_x
\end{aligned}
$$
Is the map $\delta$ ever Gateau (or Fréchet) differentiable?