Betty Skelton on the cover of Look magazine, January 1960, from her New York Times obituary:
Whether in the air or on land, Ms. Skelton, who died on Aug. 31 at the age of 85, was a celebrated daredevil who shattered speed and altitude records. She was a three-time national aerobatic women’s flight champion when she turned to race-car driving, then went on to exceed 300 m.p.h. in a jet-powered car and cross the United States in under 57 hours, breaking a record each time.
From a Associated Press article from 2008, when she was inducted to the Motorsports Hall Of Fame:
In 1959, at 33, she was the first woman to undergo NASA’s physical and psychological tests _ the same that seven original male astronauts were put through. “I complained that NASA wasn’t giving more thought to women pilots,” she said.
See also: The Mercury 13:
Cobb, already an accomplished pilot, became the first American woman (and the only one of the Mercury 13) to undergo and pass all three phases of testing. Lovelace and Cobb recruited 19 more women to take the tests, financed by the world-renowned aviatrix Jacqueline Cochran. 13 of the women passed the same tests as the Mercury 7.