A student was killed in Kenya on Monday as clashes between security forces and anti-government protesters flared after veteran opposition leader Raila Odinga called for weekly demonstrations over the spiralling cost of living.

Riot police fired tear gas and water cannon against demonstrators and Odinga's convoy as running battles erupted between stone-throwing protesters and police in parts of the capital Nairobi and at least one other city.

Senior opposition figures were also among a number of people arrested.

It was the first major unrest in the East African political and economic powerhouse since President William Ruto took office six months ago after a narrow election victory over Odinga.

In the pro-opposition town of Maseno in western Kenya, police said a student was killed after officers fired live bullets during "skirmishes" with protesters who were "pelting stones".

"William Mayange, a third-year student, was hit on the neck. He was rushed to Coptic Hospital where was pronounced dead on arrival," police said in a statement.

Odinga, 78, had called for protests against Ruto's government over economic woes faced by ordinary Kenyans and what he claims was a "stolen" election.

He upped the ante on Monday, calling for weekly protests and strikes.

"The war has begun, it will not end until Kenyans get their rights," he told supporters in Nairobi.

Earlier, police unleashed volleys of tear gas and sprays of water against his motorcade near a hotel where he had planned to hold a press conference but was forced to leave.

Odinga's spokesman Dennis Onyango said a police bullet had hit the opposition leader's vehicle, but it was not possible to independently confirm the claim.

Kenyans are reeling from the rising cost of fuel, electricity and staple foods, a slump in the value of the shilling and a record drought that has left millions hungry.

From the early morning, police used tear gas against protesters gathered near government offices in the heart of Nairobi and several other areas of the city.

About two dozen people were arrested in downtown Nairobi, mainly young people but also Senate minority leader Stewart Madzayo and his counterpart in the National Assembly Opiyo Wandayi, both members of Odinga's party.

"We came here peacefully but they tear gassed us," said Charles Oduor, 21.

"They lie to us every day. Where is the cheap maize flour they promised? Where are the jobs for the youth they promised? All they do is hire their friends."

Onyango later said Monday's protests were "a very successful outing".

"There was prior intimidation reinforced by overwhelming police presence. The people defied all that. The numbers will rise next week," he predicted.

Deputy President Rigathi Gachagua had urged organisers to call off the "mayhem and the chaos."

He said the protests had cost Kenya about two billion shillings ($15 million) in lost business.

In Nairobi's biggest slum Kibera, a bastion of Odinga support, people set tyres ablaze while police used water cannon and tear gas to disperse protesters.

Demonstrators and police also clashed in the lakeside city of Kisumu in western Kenya, another Odinga stronghold.

Nairobi police chief Adamson Bungei had said on Sunday that permission had not been granted for the demonstrations as requests had not been made in time.

Interior Minister Kithure Kindiki had also warned that anyone inciting public disorder or disturbing the peace would face prosecution.