“[Harry] Danning, a shrewd young man who can hit, has been with the Giants most of the time since 1931. This is his make-or-break season.” Hugh Bradley, New York Post – WNU syndicate, “Hugh Bradley Says: Giants Worry Less Over Flag Rivals than Own Future,” (Woodstock IL) Sentinel, March 22, 1937, p2
No earliest use given in 2009 Dickson Baseball Dictionary
NOTE: There was at least one earlier use of “make-or-break season” to refer to the financial viability of a particular minor league, but this is the first to refer to an individual player. “Make-or-break” was also used regularly in football to refer to individual games that could make or break a team’s season.