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?