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The Oxford English Dictionary has an entry related to baseball as early as 1886:

8. a. N. Amer. Of a pitcher in baseball: to cause (a batter) to strike out.
1909 in Webster's New Internat. Dict. Eng. Lang.
1912 C. Mathewson Pitching in Pinch v. 101 He fanned the next two men.
1970 Globe & Mail (Toronto) 28 Sept. 19/4 The Indian lefthander‥has fanned more than 300 hitters.

b. intr. Of a batter: to strike out.
1886 Outing July 477/2 The man who‥‘fans out' or ‘pops one up'.
1945 This Week Mag. 21 Apr. 10 He fanned in a pinch and the opposition booed.