The board game American Checkers on an 8x8 board was weakly solved in 2007 and shown to be a draw (PDF link). What is an example of a perfectly played game of checkers?
Checkers was weakly solved in 2007: it was proven that the Chinook algorithm could never lose a game. So to get a perfectly played game you could have Chinook play against itself, but I can't find the final version of Chinook online or a record of its later games. You can check out its earlier games against humans here - most of those are draws, though there's no guarantee they're "perfect" since it's a human and an earlier version of Chinook playing.