Take $V=C(X,\mathbb R)$ as the space of differentiable functions $X \to \mathbb R$ under uniform convergence. Here $X \subset \mathbb R^d$ is some compact domain. For each x is well known the map $\mathcal G:V \to C(X, \mathbb R^d)$ given by $\mathcal G(f) = \nabla f(x)$ is discontinuous. For example consider the sequence $f_n(x) = \frac{1}{n}\sin(n x)$. Clearly we have $|f_n(x)| \le 1/n $ and so $f_n \to 0$ uniformly. On the other have derivatives are $f_n'(x) = \cos(n x)$ with $f'_n(0)= 1$ and so $f'_n$ does not converge uniformly to $0$.
If we instead restrict $\mathcal G$ to the subset $F \subset X$ of continuous functions then I believe $\mathcal G:F \to C(X, \mathbb R^d)$ is continuous. But I haven't been able to make a clean proof or find a good reference so far.
For a messy proof I can use the heavy machinery in Corollary 2.3 here to see that if $f_n \to f$ uniformly then the epigraphs $\{(x,y): x \in X, y \ge f_n(x)\}$ converge to $\{(x,y): x \in X, y \ge f(x)\}$ in the sense of Kuratowski. Then using other facts I know about set convergence, I can show the gradients converge.
This all seems overblown though. I wonder is this fact well known? Could anyone provide an elementary proof or good reference please?