Nicknamed 'The Shed', the Vauxhall Corsa was the first car Smith bought after passing his driving test at 18.
Smith drove the car to University in Norwich, during his days in the National Youth theatre - as well as to some Doctor Who locations.
"While I have been travelling the universe in my TARDIS, my much-loved Vauxhall Corsa, aka The Shed, has been parked outside my parents' house. As my first ever car, it has seen lots of adventures, not to mention a fair few mishaps (hence the dents). I'll be very sad to see it go but I understand my parents want their driveway back! I'd love to find a new owner for this great little car, particularly as all the profits will go to Starlight, which is an amazing charity that brightens the lives of seriously and terminally ill children."