At approximately 7pm, the woman, who was a pedestrian, and a car collided on Bedford Road, between the junctions of Bunyan Road and Farrer Street.
Police officers responding to a separate incident were nearby, and gave medical attention while other emergency services made their way to the scene.
She was pronounced dead a short time later.
Investigators are now asking for the public’s help in identifying the woman, described as around 50 or 60-years-old, 5’6”, of Mediterranean appearance and with dark brown hair.
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Sergeant James Thorne, from the Bedfordshire, Cambridgeshire and Hertfordshire Roads Policing Unit, said: “This is obviously a very sad case and our priority is identifying this woman so that we can speak with her loved ones.
“It’s likely that she’s local to Kempston, so I would urge anyone who thinks they know someone matching the description to either check on their wellbeing, or get in touch with us.”
Anyone with any information should call 101, quoting Operation Folly.