At least two hospitals in Kenya sustained damage during protests to mark the 35th anniversary of pro-democracy rallies, witnesses and hospital officials said on Tuesday, a day after the demonstrations left 11 people dead, according to the police.

Tensions have been running high in the East African nation since the death of blogger Albert Ojwang in police custody last month brought people back onto the streets, a year after anti-government protesters stormed parliament.

In the first incident in the town of Kitengela, 34 km south of Nairobi, more than 100 unidentified attackers armed with clubs forced their way into a hospital to demand treatment for their colleagues, a nurse said.

"We had patients even in the emergency and consultation room. Some had been sent to the lab. We closed the maternity with a door grille because they wanted to go in," Lilian Nyakhurenya told Reuters.

"The maternity was full and it is always full. We were many in the shift and we ran away, and some of us had to wear the patient's uniforms."

Saningo Tatio, a clinical dermatologist, said the armed men "started damaging all the facilities" at the hospital, forcing health workers to take cover.

"The police came but the goons were so many," Tatio said, adding that the attackers also started a fire near the laboratory.

In a second incident, another group of unidentified attackers forced their way into a hospital in Nairobi, stealing equipment.

Jeff Ngami, branch manager at Bristol Hospital situated in the Embakasi neighbourhood, told Reuters that armed men stormed the premises on Monday evening and stole equipment worth over 800,000 Kenyan shillings ($6,200).

"They harassed our staff at the hospital while they were ransacking the hospital. Some of my staff were harassed, injured and traumatised."

"They also stole staff's mobile phones, optical frames and other items," he said, adding that he had reported the matter to the police.

Spokespeople for Kenyan police and the government did not respond to requests for comment.

The head of a union for health workers told reporters that officials had previously asked the government to provide them with security but their requests had gone unanswered.

"Yesterday now brought to the fore how the government has abandoned (the) security of the health worker," said Peterson Wachira, national chairman of the Kenya Union of Clinical Officers.

"We are putting the government on notice that all health facilities and especially those in the hotspots must be provided with armed security from the Kenya police."