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Maverick, season 2 episode 19 «Duel at Sundown»: air date February 1, 1959. Episode overview, season trailer and the cast of Maverick.
Bret's friend asks him to come to Sundown and convince his daughter Carrie that the local gunfighter she's in love with is a coward. The problem is that he's a gunfighter and Bret isn't.