Tom Morris Jr. was 19 when he won his third British Open in 1870.
Ko, despite a bogey at the final hole, carded 70 for a two-under 211 total, with Americans Lincicome (71) and Gerina Piller (71) the only other players under par through 54 holes.
Six others were two shots back, including Ariya Jutanugarn (68), of Thailand, who has won her past three events.
After coming from behind to win the last two major championships, the Evian Championship in September and ANA Inspiration in April, Ko is in a different position with the lead.
“Going into Evian and ANA I just had very calm thoughts, positive thoughts until the last rounds,” Ko said.
“Then I just said, ‘hey, just go out there and enjoy it. Just don’t worry about what everybody else does.’ That’s all I can do.
“I’ve got to just try my best and have fun. I can’t control what somebody else does.”
Ko began the third round one shot behind South Korean Lee Mi-rim and Canadian Brooke Henderson.
The world number one made birdies at the fifth and sixth holes to charge into the lead, but she was caught by Piller when the American made her second birdie of the day at the ninth.
Ko bogeyed the 12th, but birdied the 15th before her closing bogey, while Piller was two over on her back nine.
Jutanugarn, meanwhile, said she is not thinking about extending her run to four consecutive victories.
“Not really,” Jutanugarn said. “This one is a major and last year I missed the cut in this one, so I really wanted to make the cut.”
Also among the group two shots behind are South Korean Amy Yang, with the best round of the day of five-under 66, Swede Anna Nordqvist (69), who won the 54-hole event in New Jersey last week, Lee (73), Henderson (73) and South Korean Chella Choi (69).
(Reporting by Tim Wharnsby in Toronto; Editing by Andrew Both)
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