Let $(X_t)_{t\ge0}$ be a Hilbert space $H$ (take $H=\mathbb R^d$, for simplicity) valued time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. The latter can be considered as a contraction semigroup on the space $\mathcal B_b(H)$ of bounded Borel measurable real-valued functions on $H$ equipped with the supremum norm. Let $A$ denote the corresponding generator.
Typical examples are Lévy processes or Itō diffusions. For both of them, it holds $$\mathcal D(A)\subseteq C^2(H)\tag1.$$
Question Is there such a process $(X_t)_{t\ge0}$ for which it holds $\mathcal D(A)$ contains non-differentiable or even non-continuous functions?
Clearly, if $\kappa$ is any Markov kernel on $H$, then $$A:=\lambda\left(\kappa-\operatorname{id}_{\mathcal B_b(H)}\right)$$ for some nonnegaitve $\lambda\in\mathcal B_b(H)$ and $$\kappa_t:=e^{tA}\;\;\;\text{for }t\ge0$$ is an example of such a process (the process is a compound Poisson process). But this is not really a "natural" continuous-time process, but a discrete-time one embedded into continuous-time. So, I would like to rule out such processes from the question.