Police Scotland is appealing for witnesses to the incident which took place on Merry Street around 8.20 pm on Sunday.
The 58-year-old woman was taken to hospital where she was pronounced dead a while later.
The police van was a marked Ford Transit and was on routine duties at the time of the incident.
It did not have blue lights or sirens activated and neither officers within the van were injured.
An investigation into the circumstances is being carried out by Police Scotland’s Road Policing Unit and the incident has been referred to the Police Investigations and Review Commissioner (PIRC).
Sergeant John Tait said: “Our enquiries into this incident are ongoing and I would urge anyone who may have witnessed the collision or who has any other information to come forward.
“Anyone with information can call 101, quoting incident 3309 of 19 September.”