The force said its thoughts were with the family of Ms Armstrong, from Tibshelf in Derbyshire, following the crash, which happened in Batley Lane at about 20:00 GMT.
“Alana was just 25 – and a mum to a six-year-old boy,” Ch Supt Dave Kirby said.
“Her little boy, along with her wider family, will never get the chance to see her again, hug her again, spend Christmas together, or celebrate all those milestones.”
Police said the car that rammed the e-bike had pursued the bikes through the village of Pleasley, with the last known sighting of it in Rowthorne Lane, where it turned right on to the A617 towards New Houghton.
A man was driving the vehicle, with a woman in the passenger seat, the force added.
Officers want to hear from people who might have seen a Land Rover Discovery, thought to have been manufactured between 2004 and 2009, in and around the Pleasley area around the time of the crash.
