Mr Hammond was hurt while filming an edition of the show near Leeds.
He was reportedly trying to set a new land speed record for Britain in a jet-powered Vampire car when it overturned.
Mr Hammond was said to have been reaching speeds of up to 280mph as the accident happened.
He was taken to Leeds General Infirmary.
Firefighter Dave Ogden, who witnessed the crash, told Sky News that the 36-year-old presenter was able to speak as he was taken out of the wreckage of the car.
Inspector Mike Thompson, of North Yorkshire Police, said officers were dealing with the incident at Elvington airfield, southeast of York, where the filming was taking place.