Kim, who led by two overnight, carded six birdies to move onto 17 under par, clear of England's Georgia Hall (70) and Thailand's Moriya Jutanugarn (67).
Compatriot Inbee Park matched Michelle Wie's first-round course record 64 to move into a share of fourth.
American Ally McDonald (70) is also seven shots back on 10 under.
Park, the 29-year-old Olympic champion bidding for a second British Open title after her 2015 triumph, birdied her opening two holes, picked up three more shots before the turn and then birdied the 11th, 13th and 17th on her back nine, narrowly missing another long birdie putt on the 18th.
"I've prepared myself pretty well last week at the Scottish Open, in the bad weather and the bad wind and all sorts," she said.
"The expectation was pretty low, I just made the cut and I wasn't putting well.
"The greens were much quicker today than the last two days which helped me a lot."
The 2013 winner, Stacy Lewis, also took advantage of the relatively kind conditions to post a seven-under 65 and move up to nine under, eight shots behind.
But it is Kim who will head into the final round on Sunday enjoying a commanding lead.
"I know the expectation and I haven't made the bogeys yet," she said.
"But I think I should give myself a little bit of credit for it, and enjoy tonight. And whatever happens tomorrow happens."
Hall, from Bournemouth, carded five birdies and a terrific eagle on the fifth, but a bogey on the 16th and a double bogey on 17 saw her slip from second to tied third in her last three holes.
Fellow English duo Jodi Ewart Shadoff (71) and Mel Reid (69) are among a group of six players on eight under.
Sally Watson, the only Scot left in the tournament, dropped three shots for a round of 75 to finish level par.
South Korea's Kim In-kyung extended her lead at the Women's British Open to six shots after a third-round 66 at Kingsbarns in Fife.