After making an initial identification of the suspect, police are trying to confirm that the suspect is a 34-year-old woman from Stamford, Conn., with a history of mental health issues.
A family spokeman confirmed to ABC News that the name of the suspect is Miriam Carey.
Carey's mother, Idella Carey, told ABC News Thursday night that her daughter suffered from post-partum depression after giving birth to her daughter last August.
Idella Carey said her daughter had "no history of violence" and she didn't know why she was in Washington, D.C., on Thursday. She said she thought Carey was taking her baby to a doctor's appointment in Connecticut.
Authorities say the driver, with a 1-year-old girl in the vehicle, tried to drive through a barricade that blocks a stretch of Pennsylvania Avenue in front of the White House at 2:18 p.m. ET. The suspect fled down Pennsylvania Avenue toward Capitol Hill with Secret Service in pursuit.
The woman's car at one point had been surrounded by police cars, but she managed to escape, careening around a traffic circle and past the north side of the Capitol. Video showed officers pointing firearms at her car before she rammed a Secret Service vehicle and continued driving. Shots were fired at that location. The suspect was fatally shot a short time later about a block northeast of the Capitol.
The U.S. Capitol and the White House were immediately placed on lockdown. An alert was sent to everyone in the Capitol to shelter in place and stay away from the windows around 2:25 p.m. ET. The lockdown lasted about an hour.
Three senators told ABC News that they heard the gunfire. People standing outside the Supreme Court across the street from Congress were hurried into the court building by officers.
"We heard shots. They told us to get behind a car," said U.S. Sen. Sherrod Brown (D-OH).
No weapons were found inside the suspect's vehicle. The child was unhurt.
The motive behind the incident remains unclear, but police say they don't believe it was an accident. Officers didn't find any note in the car. The incident is believed to be isolated and not related to terrorism.
One Secret Service member and a 23-year veteran of the Capitol Police were injured. Officials said they are in good condition and expected to recover.
On Thursday night, there was a bipartisan standing ovation in the House for the officers who risked their lives to defend members.
While Capitol police officers are still working, due to the government shutdown, they will not be paid until after the budget impasse is resolved.
ABC News and The Associated Press contributed to this report.