Nakuru governor Lee Kinyanjui said 450 homes had been hit by the flood waters and safety engineers had been sent to inspect three other dams nearby.
Solai is located some 190 kilometers (120 miles) northwest of the capital of Nairobi. Local officials say neither the extent of the damage nor the death toll are yet clear.
Police spokesman Joseph Kioko said water burst through the banks of the Patel Dam, causing mass damage and sweeping away hundreds of buildings.
The Kenyan Red Cross says it has rescued 39 people so far. Local reports identified the failed structure as a privately owned earthen dam on a large farm.
UNOCHA said the flooding could get worse, with heavy rains forecast to continue in the Rift Valley and the Lake Victoria basin over the next few weeks.
Police chief Gideon Kibunjah said 36 people had been hospitalised.
The government said on Wednesday that the rain had killed 132 people and displaced 222,456 others in 32 counties since March. However, the torrential rains and floods have made it hard to reach those most in need.
The East African nation has been experiencing unprecedented flooding in recent days as torrential rains pound the country shortly after it had come out of a severe drought season marked by hunger and water scarcity. Caught between different extremes, millions of Kenyans are in dire straits. Military helicopters have been deployed to assist with relief efforts.
The extreme weather comes at a time when the country is also trying to tackle a cholera outbreak as well as an epidemic of the mosquito-linked chikungunya virus.