The chase began when the Infiniti hit a barrier on the outer security perimeter of the White House, said Ed Donovan, a Secret Service spokesman. One of the responding officers was hit by the car as it left the scene, he said.
Police said the car sped down Pennsylvania Avenue toward the Capitol, where security vehicles stopped it at Garfield Circle.
Frank Schwing, a Washington resident who was near the area, said officers "came out with their guns drawn" and approached the suspect's vehicle.
"At that point, the driver slammed into reverse, slammed into a cruiser, did a 180 (degree turn), took off, and at that point, there were a half dozen or so shots fired," apparently all by small arms from police, Schwing told CNN.
Video footage by other witnesses showed the black vehicle then careening around a nearby traffic circle with a police car in close pursuit and then headed away. Shortly afterward, the black car crashed into security barriers a few blocks, witnesses said.
Cathy Lanier, the chief of the Metropolitan Police Department, said more shots were fired after the vehicle stopped, and the woman was hit several times. She was later pronounced dead, Lanier said.