The woman struck was a pedestrian crossing the street outside of a marked crosswalk.
Photo A fleet of self-driving Uber cars in 2016.
"Our hearts go out to the victim's family", declared an Uber spokesman.
The incident is believed to be the time someone died after being hit by one of Uber's test vehicles. However, a Waymo self-driving vehicle was seen operating near the collision scene on Monday morning.
The ride-hailing firm said it was cooperating with local authorities in Arizona.
The police said Uber is assisting in the still-active investigation.
The federal government has voluntary guidelines for companies that want to test autonomous vehicles, leaving much of the regulation up to states.
The firm has such programmes in San Francisco, Pittsburgh, Toronto and the Phoenix area, which includes Tempe.
The company is devising a technology that will let its cars "talk" with pedestrians, in hopes of making such accidents less likely.
In January 2017, federal safety regulators concluded there was no defect in the Tesla Autopilot system, and that the driver should have maintained control.
California and Arizona have been particularly encouraging, hoping that companies developing autonomous technology in those states will create local jobs and facilities devoted to a promising new industry.