It happened at around lunchtime at the Fleming Way multistory.
Police were called to the area opposite Swindon Crown Court at midday (Monday) and the area was cordoned off.
Detective Sergeant Tom Straker said "At this early stage we do not believe that anyone else was involved.
"The area would have been busy at the time with shoppers and other passers-by and we would like to hear from anyone who saw a man acting suspiciously on Floor 4 of the car park around midday."
Anyone with any information can call the police non-emergency number 101.