“A rioter took advantage of the situation in the city of Najafabad and fired shots at police forces with a hunting rifle. As a result, three were wounded, and one was martyred,” said a police spokesperson.
The names of the officers have not been released.
The death is the first report of a police or government fatality since demonstrations began last week.
At least 12 protesters are said to have been killed– 10 of them overnight Sunday – in the protests, which started on Thursday in Iran’s second city, Mashhad.
Protests were breaking out again on Monday night in the capital Tehran, according to local media.
The Fars news agency showed a burning car, but reports on social media claimed it involved relatively small groups chanting anti-regime slogans.
Unemployment and inflation is very high in Iran, and a lifting of sanctions in January 2016 has failed to deliver the economic boom many hoped for.
The demonstrations are believed to have been sparked by an increase in egg and poultry prices by as much as 40%.
State television claimed that on Sunday night “armed protesters tried to take over some police stations and military bases”.
The protests are the boldest challenge to Iran’s leadership for years and authorities say more than 400 people have been arrested.
The situation has drawn comment from many world leaders.
Israel’s prime minister, Benjamin Netanyahu, said on Twitter: “The Iranian regime tries desperately to sow hate between Iranians and Israelis. They won’t succeed.
“When this regime finally falls – and one day it will – Iranians and Israelis will be great friends once again. I wish the Iranian people success in their noble quest for freedom.”
The Iranian regime tries desperately to sow hate between Iranians and Israelis. They won’t succeed. When this regime finally falls – and one day it will – Iranians and Israelis will be great friends once again. I wish the Iranian people success in their noble quest for freedom. pic.twitter.com/kk8wTYmhnz
— Benjamin Netanyahu (@netanyahu) January 1, 2018
US President Donald Trump has also intervened, demanding change and referring to the nuclear deal agreed with Iran by Barack Obama in 2015.
Mr Trump tweeted: “Iran is failing at every level despite the terrible deal made with them by the Obama Administration.
“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!”
The post followed a series of tweetssent by Mr Trump warning Tehran to respect freedom of speech.
Iranian President Hassan Rouhani has called for calmin a televised address to the nation.
He said people had a right to protest as long as it did not lead to violence.
President Rouhani said his regime must allow “space for legal criticism” – but that “criticism is different to violence and destroying public property”.
“The government will show no tolerance for those who damage public properties, violate public order and create unrest in the society,” he warned.
Iran has blocked access to Instagram and the messaging app Telegram to try to damp down the protests. Both apps were used by activists to organise the demonstrations and share images and information.
These are the most serious demonstrations since 2009, when Iran saw eight months of civil unrest caused by its disputed presidential election.
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However, there have also been pro-regime support on the streets.
On Saturday, tens of thousands marched in cities across Iran in a show of support for the government and Supreme Leader Ayatollah Ali Khamenei.