The breach happened on Wednesday near the town of Solai, 190km (120 miles) north-west of the capital, Nairobi.
At least 200 people have been rendered homeless.
The dead are thought to include children and women trapped in mud. The Kenyan Red Cross says it has rescued some 40 people so far.
Local officials say the full extent of the damage is not yet clear. There are fears the death toll could rise as the search-and-rescue operation continues.
The heavy rains in Kenya and other regional states come after a severe drought which left millions of people in need of food aid.
Nakuru County Police Chief Gideon Kibunjah said that 32 bodies had been recovered so far.
This included 11 bodies, mostly of women and children, covered with mud at a coffee plantation, an unnamed police officer told AFP news agency.
It seemed that they had been fleeing, but “could not make it due to the force and speed of the water from the flooded dam”, the officer added.
The Patel dam, located on private farmland, and reportedly used for irrigation and fish farming, broke its walls and swept away hundreds of homes downstream.
“The water has caused huge destruction of both life and property. The extent of the damage has yet to be ascertained,” said Lee Kinyajui, Governor of Nakuru County
Survivor Veronica Wanjiku Ngigi, 67, told Reuters news agency that she was at home brewing tea when her son’s wife rushed in to say they needed to get to higher ground as the dam had burst.
“It was a sea of water. My neighbour was killed when the water smashed through the wall of his house. He was blind so he could not run. They found his body in the morning,” she was quoted as saying.
Local leaders are now seeking to find out whether the farmer was licensed to erect those dams, amid concerns about the condition of the remaining two which are also said to be full, reports the BBC’s Ferdinand Omondi in the capital, Nairobi.
The toll now brings to 162 the number of people who have died countrywide as a result of heavy rains since March, according to official statistics.
More than 220,000 people have also had their homes destroyed. -BBC
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