Let $X$ be a Banach space, and let $f : X \to \mathbb{R}\cup\{+\infty\}$ be proper convex. Assume that $f$ is locally bounded above at $x \in X$, so that $f$ is locally Lipschitz at $x$. Assume also that $\partial f(x) = \{y\}$ is a singleton, so that $f$ is Gâteaux differentiable at $x$.
My question: Is the condition of local Lipschitz continuity strong enough to give Fréchet differentiability?