Secret Service Spokeswoman Ann Lomax said the president was "never in any danger." White House Spokesman Trent Duffy said Bush had just got into his limousine.
Bush had completed a speech to campaign supporters inside the DeSoto Civic Center at the time of the crash, according to Lomax.
Witnesses said a woman and three children were in the car, a 4-door Toyota Camry with at least one flat tire. She drove down a street at high speed pursued by police and sheriff's deputies, they said.
The car passed the presidential motorcade, took a sharp left, jumped a curb, and barreled across the grass outside the center before hitting the building near one of its entrances, witnesses said.
Bush had just got into his limousine at the time of the crash, according to White House spokesman Trent Duffy.
The president "was never in any danger," Lomax said. The car blew out some tires when it jumped the curb, according to Lomax.
The car hit the building near its upper loading dock area, Lomax said. The president's motorcade was in the lower loading deck area, she said.
Authorities surrounded the car, and no shots were fired, Lomax said.
The woman was arrested immediately after the incident, which took place about 10:30 a.m. (11:30 a.m. EST). The Secret Service is leading an investigation into the crash, according to Lomax.
Video from the scene showed four officials moving the woman, with her arms behind her back, into a white unmarked car.
Chief David Mitchell, spokesman for the DeSoto County Sheriff's Department, would not say why the woman was being pursued. He said the children in the vehicle are with family members and "being well taken care of."
Local and federal officials are conducting a joint investigation, Mitchell said. It will remain a joint investigation until any charges are filed, he said.
Bush left the center shortly after the crash and headed to his next campaign stop in Paducah, Kentucky.