Witnesses said the vehicle was speeding.
Kimilili sub-county police commander Nelson Marwa said they were investigating the incident.
He called for calm among residents as the probe continues.
Nabudwa is a Ugandan who was married in Kimilili. Her death caused a sombre mood in the area.
Her husband James Thuku Ndicho said the family was in deep sorrow.
Ndicho told reporters that he and his wife had taken breakfast together that fateful day before she left for work.
He was later called at around 7 pm and informed that his wife had been knocked down by a police car.
"I rushed to the scene and found my wife dead. I was so saddened that I couldn't speak a word. I am appealing to relevant authorities to help us get justice for my dear wife," a tearful Ndicho said.
Most businesses in Bungoma close early so residents can beat the 7 pm curfew imposed in the area. Bungoma is among 13 lake region counties where Covid-19 restrictions were escalated two weeks ago to curb the spread of the Delta variant of the disease.
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