Let S be the set of those α∈ω₂ that have cofinality ω₁. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length ω₁, but not the Banach-Mazur game of length ω+1 (both games starting with S).
"Winning the pressing down game but not Banach-Mazur." J. Symbolic Logic 72 (4) 1323 - 1335, December 2007. https://doi.org/10.2178/jsl/1203350789