Is the molecular Berry phase an artifact of the Born-Oppenheimer approximation?

We demonstrate that the molecular Berry phase and the corresponding non-analyticity in the electronic Born-Oppenheimer wavefunction is, in general, not a true topological feature of the exact solution of the full electron-nuclear Schr\"odinger equation. For a numerically exactly solvable model we show that a non-analyticity, and the associated geometric phase, only appear in the limit of infinite nuclear mass, while a perfectly smooth behavior is found for any finite nuclear mass.