Sources told NBC news the suspect has been identified as Miriam Carey, 34, of Stamford, Conn., a dental hygienist with a history of mental illness.
Washington Police Chief Cathy Lanier told reporters at a Capitol Hill news conference investigators would not "answer any questions about the suspect at this point," but she added, "Right now the suspect has been pronounced [dead]."
A 1-year-old child in the woman's car -- video showed her to be a girl -- "is in good condition and in protective custody," Lanier said.
The chief said "shots were fired" by police at two points, at Garfield Circle west of the Capitol and near the U.S. Supreme Court, when the woman's car crashed.
All the shots apparently were fired by police.
Lanier said two officers, a uniformed Secret Service guard at the White House and a Capitol Police officer, were injured, but not seriously.
Secret Service spokesman Ed Donovan said before the chase the woman's car struck a temporary barrier and a uniformed guard near the White House. He said the identity of the woman is "not being released at this point."
Capitol Hill Police Chief Kim Dine said the incident "appears to be an isolated singular matter at this point with no nexus to terrorism."
He also refused to release any information about the suspect, but said her toddler was in good shape and taken to a hospital.
The woman suspect in the high speed car chase appears to have been driving a car registered in Connecticut.
The Connecticut plates on the woman's vehicle were clearly visible on video of the incident.
Video on several news channels from Alhurra, an Arabic network based in Washington, showed officers surrounding the car near the Botanical Gardens, west of the U.S. Capitol. The stopped vehicle slams back into a police vehicle, injuring an officer. At that point police around the car began firing into the vehicle, apparently unaware a small child was inside with the woman.
The woman drove away at high speed, circling the Peace Memorial, before being stopped at Second Street and Massachusetts Avenue.
The Capitol was placed on lockdown and business was suspended during the incident. After the all-clear was given, the House went back into session. The Senate was in recess until 10:30 a.m. Friday.
The White House said President Barack Obama was briefed on the matter and White House staff were in touch with law enforcement and monitoring the incident.
The Capitol already is closed to visitor tours due to the federal government shutdown.