The Ken Johnson answer is correct now. However, prior to 1996, a no-hitter was defined as “a game in which a team has no hits through nine innings.” But in May of that year Major League Baseball (Elias Sports Bureau) revised the official definition of a no-hitter as “a game of at least nine full innings which ends with a team having no hits.” “Official” status was made retroactive. There were 12 games that lost no-hit status because they went extra innings and the first hit was made after the 9th. In nine of these games, the "no-hit" team ended up winning the game.
May 5, 1901 - Earl Moore, CLE vs CHI AL
August 1, 1906 - Harry McIntire, BKN at PIT NL
April 15, 1909 - Leon Ames, NY vs BKN NL
August 30, 1910 - Tom Hughes, NY vs CLE AL
May 2, 1917 - Hippo Vaughn, CHI vs CIN NLa
September 18, 1934 - Buck Newsome, STL vs BOS AL
May 26, 1956 - Johnny Klippstein, Hershell Freeman, & Joe Black, CIN at MIL NL
May 26, 1959 - Harvey Haddix, MIL vs PIT NLb
June 14, 1965 - Jim Maloney, CIN vs NY NL
July 26, 1991 - Mark Gardner, MON at LA NL
June 3, 1995 - Pedro Martinez, MON at LA NLc
a - Part of the fabled 'double no-hitter', as Fred Tooney of the Reds completed a 10-inning no-hitter.
b - Haddix had a perfect game through 12 innings, a major league record. He lost the game on an error, a walk, and a hit in the 13th.
c - Martinez had a perfect game through 9 innings, lost the game on two hits in the 10th.
In addition, there have been two games in which the pitcher did not allow any hits while losing on the road, meaning there was no home ninth inning played. These games are not “official” no-hitters under the 1996 ruling.
July 1, 1990 - Andy Hawkins NY at CHI AL
April 12, 1992 - Matt Young BOS at CLE AL