Free evolution conserves analyticity but what if the initial state $\Psi_0$ is real-analytic but the acting Hamiltonian includes a non-analytic or even singular potential, like the Coulomb central potential $V(x) = -1/|x|$? One would suppose that in this case, analyticity gets destroyed, at least for almost all times $t \neq 0$.
If one scans the literature a positive analyticity result can be found in [1] for analytic potentials in one space dimension. Counter-examples where analyticity is lost when potentials are not that nice are hard to find, there is a statement of yuggib on StackExchange Physics that evolution with a Coulomb potential already kills continuous differentiability, but without reference or proof. And there is Theorem 1.2 in [2] that states that the fundamental solution to the time-dependent Schrödinger equation is nowhere $\mathcal{C}^1$ for a certain class of potentials. But this class does not include Coulomb potentials and the properties of the fundamental solution do not necessarily carry over to the solution itself. In [3] we read:
On the other hand, if $V$ is not smooth, e.g., if $V$ is the Coulomb potential in dimension three, the singularities of $V$ create those of the FDS and $E(t, x, y)$ [the fundamental solution] is not smooth everywhere. However, the strong dissipation property of the free propagator $\mathrm{e}^{-\mathrm{itH_0}}$ moderates the singularities and we expect that $E(t, x, y)$ is bounded and continuous for $t\neq 0$ if $V$ is bounded at infinity in a suitable norm and is not too singular locally (see Simon [14] [ref. [4]] who conjectures that this is true if $V$ is of Kato class).
The author goes on and shows boundedness and continuity for the fundamental solution for a class that includes the Coulomb potential in dimension three. Finally one should note that for a Hamiltonian with Coulomb potentials and even including Coulomb interactions between multiple particles, it is known that an eigenstate is analytic away from the potential sigularities and has $|x|$-formed cusps at the location of the singularities [5].