* Islamists undeterred by death toll, promise more protests
* Ministry accuses Brotherhood members of terrorist acts
* Government says 173 died Friday, over 1,000 arrested (Recasts, adds quotes)
By Issam Abdallah and Yasmine Saleh
CAIRO, Aug 17 (Reuters) - Supporters of deposed President Mohamed Mursi fought a gunbattle with security forces in a Cairo mosque on Saturday, while Egypt's army-backed government, facing deepening chaos, considered banning his Muslim Brotherhood group.
Three Reuters witnesses saw gunmen shoot from a window of the al-Fath mosque, where Brotherhood followers sheltered during ferocious confrontations in the heart of Cairo on Friday.
Another gunman was shown on television shooting from the mosque's minaret and soldiers outside returning fire. It was not clear if anyone died in the latest clash - the fourth day of violence in Egypt, which has killed almost 800 people.
With anger rising on all sides, Prime Minister Hazem el-Beblawi proposed disbanding the Brotherhood, raising the stakes in a bloody struggle between the state and Islamists for control of the Arab world's most populous nation.
"We are not facing political divisions, we are facing a war being waged by extremists developing daily into terrorism," presidential political adviser Mostafa Hegazy told reporters.
If Beblawi's proposal to disband the Brotherhood is acted on, it would force the group underground and could herald large-scale arrests against its members placed outside the law.
Many Western allies have denounced the killings, including the United States, alarmed by the chaos in a country which has a strategic peace treaty with Israel and operates the Suez Canal, a major artery of world trade.
However, Saudi Arabia threw its weight behind the army-backed government on Friday, accusing its old foes in the Muslim Brotherhood of trying to destabilise Egypt.
The health ministry said 173 people died in clashes across Egypt on Friday, including 95 in central Cairo, after the Brotherhood called a "Day of Rage" to denounce a crackdown on its followers on Wednesday that killed at least 578 people.
Fifty-seven policemen died over the past three days, the interior ministry said.
Among those killed on Friday was a son of Muslim Brotherhood leader Mohamed Badie, shot dead close to al-Fath mosque, which was rapidly transformed into a makeshift morgue and a refuge for hundreds of Mursi's supporters, looking to escape the bloodshed.
The building was surrounded overnight and police fired volleys of tear gas into the carpeted prayer hall in the early afternoon, filling the hall with billowing white smoke and leaving those inside gasping for breath.
Soon afterwards gunshots rang out from both sides.
Egyptian authorities said they had rounded up more than 1,000 Islamists after Friday's protests, showing one handcuffed man on television with an automatic gun on his lap.
Security sources said Mohamed Al-Zawahiri, the brother of al Qaeda leader Ayman Al-Zawahiri, had also been detained.
"Friday was a very bad, ugly day. There were attacks on police stations, ministries. The situation is very bad," the prime minister told reporters. "There will be no reconciliation with those whose hands have been stained with blood and who turned weapons against the state and its institutions."
The Brotherhood was officially dissolved by Egypt's military rulers in 1954, but registered itself as a non-governmental organisation in March in a response to a court case brought by opponents of the group who were contesting its legality.
Founded in 1928, the movement has deep roots in the provinces and has a legally registered political arm - the Freedom and Justice Party - which was set up in 2011 after unrest that led to the downfall of the autocratic Hosni Mubarak.
The Brotherhood won all five elections that followed the toppling of Mubarak, and Mursi governed the country for a year until he was undermined by mammoth rallies called by critics who denounced his rule as incompetent and partisan.
Army chief Abdel Fattah al-Sisi says he removed Mursi from office on July 3 to protect the country from possible civil war.
Brotherhood spokesman Gehad El-Haddad said the police had started to arrest the sons and daughters of the organisation's leadership in an effort to gain leverage.
Despite the bloodshed, the Islamist group has urged its supporters to take to the streets every day this coming week, but there was no sign of large rallies by Saturday afternoon.
"Our rejection of the coup regime has become an Islamic, national and ethical obligation that we can never abandon," said the Brotherhood, which has accused the military of plotting the downfall of Mursi to regain the levers of power.
Iraq, wracked by its own civil strife, urged restraint.
"We were shocked by the huge number of casualties that resulted from breaking up the protests and by the excessive use of power," Iraqi Foreign Minister Hoshiyar Zebari told U.S-based al Hurra television.
Worryingly for the Egyptian army, violence was reported across the country on Friday, with deaths reported in at least eight cities and towns, suggesting it might struggle to impose control on the vast, largely desert state.
The government said 12 churches had been attacked and burned on Friday, blaming the Islamists for the destruction.
Foreign journalists in Cairo said they faced regular harassment as they tried to report on the clashes, with a number detained by police and civilian vigilante groups unhappy with coverage of the disturbances. (Additional reporting by Tom Perry, Michael Georgy, Tom Finn, Mohamed Abdellah, Ahmed Tolba and Omar Fahmy, Writing by Crispian Balmer; Editing by David Stamp)