Bramall Lane was built in 1855.A much contested claim by a few clubs I believe, highly controversial stuff, but I'm going with Great Yarmouth?
Maybe Bramall Lane but I think that was formerly a cricket ground?
From the Wiki;
"Deepdale is now football's oldest ground in terms of continuous use by a league club. Preston North End was a founder member of the Football League in 1888"
Initially mostly cricket, but was also used for football (e.g. final of Youdan Cup, 1867, the first ever football tournament, won by Hallam FC).
Sheffield Utd were founded in 1889 with Bramall Lane as their home ground.
Cricket use ended 1974/5, when the South Stand was built over what had been the square.
Hallam FC have been playing at their original ground since 1860, but they didn't make the league when it was set up and have remained a small local club ever since. I used to walk past the ground to/from school every day.