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Wiltshire Police have released two CCTV pictures of the car park at Sainsbury's in Chippenham, where a body was found after a fire there this week.
Police have told Heart they are confident they know who the dead woman is and her family have been told.
They say they are continuing to investigate the circumstances of this untimely and sudden death on behalf of H M Coroner for Wiltshire but are no longer treating the death as suspicious.
Her body was discovered underneath debris once the flames had been put out early on Tuesday morning.
Detectives are now hoping people pictured in the CCTV that morning might have information to help the investigation.
One image (left) is of a man who walked through the car park with a bike at 1.54am.
Another (above right) is of a small dark coloured hatchback, whose occupants were seen driving through the car park in the direction of McDonalds at 2.48am.
The fire in part of the supermarket car park was reported around 3am by a police dog handler, who was on a routine patrol.
The body was removed and then taken to Flax Bourton mortuary near Bristol, where a post-mortem was carried out the same day.
The car park was cordoned off that day and the supermarket was also closed, but both were reopened on Wednesday.
Wiltshire Police can be contacted on 101 or via Crimestoppers, anonymously if preferred, on 0800 555 111.