Winning the pressing down game but not Banach Mazur
  Let $S$ be the set of those $\alpha\in\omega_2$ that have cofinality $\omega_1$. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length $\omega_1$, but not the Banach Mazur game of length $\omega+1$ (both games starting with $S$). 
Jakob Kellner, Matti Pauna, Saharon Shelah