The dam failed near the town of Solai, Nakuru County, along the Rift Valley, around 120 miles northwest of the capital Nairobi.
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"We assure residents that we will do our best to take affected families to safety and help them get medical attention", he said in a statement.
The Kenyan Red Cross tweeted that it had rescued 39 people.
A local television station said 20 bodies had been recovered so far.
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Nakuru county governor Lee Kinyanjui told Reuters as many as 2,000 people had been affected by the rupturing of the dam on Wednesday night.
"The extent of the damage has yet to be ascertained".
A search and rescue operation led by police was set up and includes officers from the National Youth Service (NYS) as well as members of the public.
Police officers speaking from the scene say the private Patel dam, used for irrigation and fish farming, may have lacked a proper outlet.
A disaster management team from the Nakuru County Government is also evacuating survivors to the Bahati Sub County and Nakuru Level Five hospitals.
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The Patel dam belongs to a farmer who also has two other reservoirs on their land.
Even before this week's dam-burst, rains had killed 132 people and displaced 222,000 in Kenya alone, according to the government.
"The county has dispatched ambulances to the scene to aid in evacuation of victims", he said.
"We urge residents to continue to be vigilant due to flooding caused by heavy rains experienced in the county and other parts of the country", he said.
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