A protester shot and killed a policeman during demonstrations in Iran, police said, the first reported fatality among security forces struggling to contain unrest challenging the clerical leadership.

Demonstrations continued for a fifth day, after 13 people were reported to have been killed yesterday in the worst wave of unrest since 2009 when huge crowds took to the streets to condemn the re-election of then-president Mahmoud Ahmadinejad.

The protests put pressure on the clerical leaders in power since the 1979 Islamic Revolution, as well as President Hassan Rouhani who made a televised call for calm, saying Iranians had the right to criticise but must not cause unrest.

The killing of the police officer happened when a demonstrator opened fire with a hunting rifle in the central city of Najafabad, a police spokesman was quoted as saying by state TV.

Three policemen were wounded.

It was not clear when the incident took place.

Hundreds of people have been arrested, according to officials and social media.

Online video showed police in the capital Tehran firing water cannon to disperse demonstrators, in footage said to have been filmed yesterday.

Protests against economic hardships and alleged corruption erupted in Iran's second city of Mashhad on Thursday and escalated across the country into calls for the religious establishment to step down.

Some of the anger was directed at Ayatollah Ali Khamenei, breaking a taboo surrounding the man who has been supreme leader of Iran since 1989.

"The government will show no tolerance for those who damage public property, violate public order and create unrest in society," Mr Rouhani said in his address yesterday.

US President Donald Trump also tweeted: "The great Iranian people have been repressed for many years. They are hungry for food & for freedom. Along with human rights, the wealth of Iran is being looted. TIME FOR CHANGE!"

Israeli Prime Minister Benjamin Netanyahu praised the "brave Iranians" taking to streets to protest at a regime that "wastes tens of billions of dollars spreading hate".

"I wish the Iranian people success in their noble quest for freedom," he said in a video posted on his Facebook page.

German Foreign Minister Sigmar Gabriel urged "all sides (to)refrain from violent actions".