The set of subdifferentials on a single point forms a convex set (can be easily shown by using half-space arguments). However, the collection of subdifferentials on a set might not form a convex set, even if the function and set are convex.
To make it more concrete, let $f:\mathbb R^n \to \mathbb R$ be a convex function, and $E \subset \mathbb R^n$ be an open convex set. Is there a counterexample showing that $\partial f(E):= \{g | g \in \partial f(x) \text { for some $x \in E$}\}$ is not a convex set?
There is an obvious counterexample if convexity condition on $E$ is dropped, by considering $f(x) = |x|$ on $\mathbb R$ with $E = \mathbb R \setminus \{0\}$. However, with convexity condition on a domain, creating a counterexample seems extremely difficult.