A total of 87 players have scored 139 competitive hat-tricks, but here's the list of players who have scored four or more goals in a game.
Alex Higgins is the only player to score 5 in a game, and it was in the same 14-0 win over Clapton that Tinsley Lindley scored his only 4-goal game.

Thomas Peacock scored 4 goals 4 times, and the final three games were all within 6 weeks of other during the lead-up to Christmas 1935.
Season Date Game Scorer Goals Opponent H/A Score Result Ground
1890-91Sat Jan 17, 1891FA CupAlex Higgins5ClaptonA14-0WinSpotted Dog Ground
1882-83Sat Dec 02, 1882FA CupSam Widdowson4Sheffield HeeleyH7-2WinTown Ground
1885-86Sat Oct 31, 1885FA CupGeorge Tutin4MellorsH6-2WinGregory Ground
1890-91Sat Jan 17, 1891FA CupTinsley Lindley4ClaptonA14-0WinSpotted Dog Ground
1893-94Sat Sep 02, 1893League Tier 1James Collins4Wolverhampton WanderersH7-1WinTown Ground
1907-08Sat Nov 09, 1907League Tier 1Enoch West4SunderlandH4-1Win
1912-13Sat Jan 25, 1913League Tier 2Thomas Gibson4BurnleyA5-3WinTurf Moor
1933-34Sat Dec 23, 1933League Tier 2Thomas Peacock4Port ValeH6-1Win
1935-36Sat Nov 09, 1935League Tier 2Thomas Peacock4BarnsleyH6-0Win
Sat Nov 23, 1935League Tier 2Thomas Peacock4Port ValeH9-2Win
Thu Dec 26, 1935League Tier 2Thomas Peacock4Doncaster RoversH6-2Win
1950-51Sat Nov 18, 1950League Tier 3Tommy Capel4GillinghamH9-2Win
1952-53Fri Dec 26, 1952League Tier 2Wally Ardron4Hull CityH4-1Win
1977-78Tue Oct 04, 1977League Tier 1Peter Withe4Ipswich TownH4-0Win
1988-89Wed Jan 18, 1989League CupLee Chapman4Queens Park RangersH5-2Win
2002-03Sat Feb 22, 2003League Tier 2Marlon Harewood4Stoke CityH6-0Win
2011-12Tue Mar 20, 2012League Tier 2Garath McCleary4Leeds UnitedA7-3WinElland Road