Wicketkeeper Batty's catches came as Surrey bowled out Kent for 239, the visitors following on 240 behind.
His efforts behind the stumps followed on from Batty's 129 in Surrey's first innings 479 all out.
Batty is the third Englishman and fourth keeper overall, following on from Australia's Wally Grout, to have taken eight catches in an innings.
But Kent rubbed some of the shine off Surrey's day by making 254 for four in their second innings - a narrow lead of 14 as the sides' headed into Friday's final day of four. Kent and England batsman Robert Key was unbeaten on 134 not out.
The record for most catches taken in a first-class innings by an outfielder is seven, held jointly by Surrey's Micky Stewart against Northamptonshire at Northampton in 1957 and Tony Brown for Gloucestershire against Nottinghamshire in 1966.
Most wicket-keeping catches in a first-class innings
8 Wally Grout, Queensland v Western Australia at Brisbane, 1959-60
8 David East, Essex v Somerset at Taunton, 1985
8 Steve Marsh, Kent v Middlesex at Lord's, 1991
8 Jon Batty, Surrey v Kent at The Oval, 2004