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Timewise, the longest game was 9 hours 52 minutes.
Inning wise, the longest game was 26 innings on May 1, 1920 between Brooklyn and Boston.
You can see more baseball recods here: http://www.baseball-almanac.com/recbooks/rb_gmlg.shtml
I appologize, you asked what the longest baseball game played was, so I gave you the answer and the link above.
If you're looking for a game specifically around 1960 for a NY team, and not longest game on record, I am not finding anything.
I will opt out of this question and let another expert find the answer for you.
Perhaps this is what you were looking for. The first extra innings game ever played at Shea Stadium was May 31, 1964. It went 23 innings; it might have seemed like the longest game, because it was the second game of a doubleheader that day.
I'm so pleased it was the right one! The final score was San Francisco Giants 5, Mets 3. I've included a link to the box score for the game: http://www.baseball-almanac.com/box-scores/boxscore.php?boxid=196405311NYN
Thanks for the question!