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.
Chinook can definitely still lose a game. When the proof was announced in 2007, it was unbeatable in 28 of the 156 3-move ballots and, theoretically, beatable in the other 128. It still is. In fact, in a match of indefinite length against a genuine oracle it would definitely lose. Currently the KingsRow program (Ed Gilbert) is stronger than Chinook - it's phenomenal - though still far from perfect.
In a 40-game match between Chinook and the best 'version' of the late Dr Marion Tinsley, it would be a very close call - Marion could still win, although Chinook would be the favourite by a very tiny margin. Richard Pask Author of Complete Checkers and Checkers for the Novice