By Adam Brookes and Crispin Thorold
BBC news correspondents in Baghdad
Iraqi Prime Minister Nouri Maliki has extended by 10 days a deadline for Shia militiamen fighting troops in the southern city of Basra to hand over their weapons.
Dozens of people have died in the latest outbreak of violence
Militiamen - in particular those of the Mehdi Army, loyal to the cleric Moqtada Sadr - show no signs of giving up and fighting continues.
Mehdi militiamen are holding key points around Basra, say local sources, and are harassing Iraqi troops from alleyways and back streets, where armoured vehicles find it hard to manoeuvre.
Mr Maliki has not yet reached his stated objective of clearing the city of militias and imposing law and order.
Local members of parliament who belong to Mr Sadr's political organisation said in a news conference that they were prepared to negotiate, but that their militiamen had the right to act in self-defence.
"Whoever points a gun at us, we have the right to kill them," said one.
Sadr supporters say the crackdown is politically motivated
Basra appears to be paralysed. Local residents say supplies of food and fuel are running low.
A phone call came to the BBC bureau from Timimiyah district - an area of Basra still held by the Mehdi Army.
Water supplies have been cut off because of the fighting, and people have not been able to go out to buy food.
The caller said his father had had a heart attack and had no medication, and that the Iraqi army had raided the local market and taken all the food.
In Baghdad, the Mehdi Army militiamen were back on the streets, armed and defiant.
They have clashed with Iraqi forces in and around the Sadr City neighbourhood, the Mehdi Army's stronghold.
Police sources report 36 dead and 147 wounded.
Eyewitnesses reported seeing freshly laid roadside bombs at the entrances to Sadr City.
They told of gunmen stopping people on the street and insisting they stay away from work in the name of civil disobedience.
Barricades have gone up, and the Iraqi police are reported to have closed bridges leading into Sadr City.
Elements of the Mehdi militia are still firing rockets into central Baghdad.
Several salvoes have gone into the Green Zone, which houses the Iraqi government.
When they fall short, the rockets plummet into crowded neighbourhoods and kill Iraqi civilians.
Two people died when one landed next to the foreign ministry, on a bus terminal.
Another died in the Kerrada district, and another in the district of Ur.
The Shia who live in Sadr City took to the streets in their thousands on Thursday morning for a demonstration in support of Moqtada Sadr.
We spoke to a demonstrator by phone. He said Mr Sadr's followers were "peaceful" and would obey their leader's admonition to observe the ceasefire of the past year.
"We are very patient," he said, "but if the government does not respond to our demands, something bad will happen."
His demands: the prime minister must resign; foreign troops must leave Iraq; the operation in Basra must be halted.
An ambitious agenda, to say the least - and one likely to be ignored by the government.
We have a series of fragmentary reports of clashes between Mehdi militiamen and the armed forces in other towns, too.
Hilla, Kut and Diwaniyah have all seen fighting. The scale of the clashes and casualty figures are hard to pin down.
The political wing of the Sadrist movement is attempting to sound conciliatory, calling on the prime minister to leave Basra and enter negotiations.
The armed wing of the movement appears to be gearing up for a fight.
The prime minister, the Americans and the British say the Basra operation is a legitimate attempt by the state to exercise its authority and to cleanse Basra of militia activity.
The US line is that the Sadrist movement is a legitimate political entity, and those resisting in Basra are "special groups" - out-of-control, dangerous elements who are being stirred up by Iran.
Mr Maliki's ultimatum is a risky gamble
The Americans applaud the Basra operation.
Certainly, a stable Basra rid of militia violence and corruption is in the interest of the central government and the coalition.
But to many Iraqis, this appears to be a risky confrontation.
At stake are power and resources in the south of the country.
Many point to the fact that Mr Maliki's own political base lies with Shia parties who view the Sadrists with suspicion.
They tell us that the prime minister wishes to weaken the Sadrist movement before local elections due later this year, for fear the Sadrists will win.
Iraq appears to be at a dangerous moment.
If the Mehdi gunmen ignore the Saturday deadline, the prime minister will have to decide whether he pursues and extends his military operations in Basra.
If he backs down, and lets the deadline pass, it will seriously damage his credibility.
But if the Mehdi Army decides to fight back in earnest, the prospect of a frightening increase in violence looms, in Basra and around the country.
US General David Petraeus has always maintained that the increased security across Iraq in recent months is "fragile and reversible".
This moment is worrying evidence of that assertion.