The answer seems to be negative, proved by Gabriel Debs in 1985:

 - <cite authors="Debs, Gabriel">_Debs, Gabriel_, [**Stratégies gagnantes dans certains jeux topologiques (Winning strategies in certain topological games)**](http://dx.doi.org/10.4064/fm-126-1-93-105), Fundam. Math. 126, 93-105 (1985). [ZBL0587.54033](https://zbmath.org/?q=an:0587.54033).</cite>
https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/126/1/104687/strategies-gagnantes-dans-certains-jeux-topologiques

The paper is in French, but here is the [MathScinet review](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=55795&sort=Newest&vfpref=html&r=35&mx-pid=817083):

 > A Banach-Mazur game on a topological space is a two-person infinite game in which a move of each player is to choose an open set contained in that previously picked by the opponent. The first player wins if the resulting descending sequence of open sets has empty intersection. Otherwise the second player wins. It is proved that if the second player has a winning strategy depending on the entire sequence of previous moves of both players then he also has a winning strategy depending on the last two moves played (his own last move and his opponent's). This was independently discovered by F. Galvin and R. Telgarskybut the author proves it for a more general class of games. The second result is the construction of a completely regular space on which the second player has a winning strategy depending on the last two moves of the opponent but does not have a winning strategy depending only on the opponent's last move.
>
> Reviewed by Andrzej Pelc