[Ed. note: This story has been updated with a more complete version of the account.]
By Tom Perry and Alexander Dziadosz
CAIRO (Reuters) - At least three protesters were shot dead on Friday outside the Republican Guard barracks in Cairo where deposed President Mohamed Mursi is being held, security sources said, as angry Islamist supporters confronted troops across the country.
Thousands of people marched across the country in what Mursi's Muslim Brotherhood movement called a "Friday of Rage" to protest against his ouster and an interim government set up to prepare for fresh elections.
Egypt's first freely elected president was toppled on Wednesday in what his Islamist supporters call a military coup.
Mursi supporters in Cairo were hit by shotgun pellets after a crowd of several hundred people marched towards the barracks where Mursi is being held. Reuters photographers took pictures of at least one dead young man and several severely wounded being carried from the scene.
The army denied blame for the shootings. An army spokesman said troops did not open fire on the demonstrators and soldiers used only blank rounds and teargas to control the crowd. It was unclear whether security forces units other than army troops were also present.
Later, tens of thousands of cheering Islamists gathered near a mosque in a Cairo suburb where they were addressed by Brotherhood leader Mohamed Badie, free to address them despite reports on Thursday that he had been arrested.
In a fiery speech, he vowed to "complete the revolution", and repeatedly referred to Mursi as the president.
"To the great Egyptian army, I say 'Allahu Akbar' (God is great) ... I say ... we will sacrifice," he shouted as a military helicopter hovered low overhead.
He urged the army not to fire on its own citizens, and added: "Our bare chests are stronger than bullets."
Continued violence would alarm the United States. Washington has so far avoided referring to the army's removal of Mursi as a "coup", a word that under U.S. law would require a halt to its $1.5 billion in annual aid. Mursi's opponents also say it was not a coup but an intervention to impose the "people's will".
Egypt has been in turmoil since the fall of Hosni Mubarak in the "Arab Spring" revolutions that swept the region in 2011.
Several dozen people have been killed in the last month of unrest, during which huge rallies in Cairo and other cities called for Mursi's resignation amid anger over economic stagnation and perceptions of a Brotherhood power grab.
His overthrow on Wednesday was greeted with wild scenes of celebration involving millions of people, but also infuriated his supporters who fear a return to the suppression of Islamists they endured under generations of military rule.
Mohamed Ezzat, 35, who said he was a brotherhood member, said protesters would stage a sit-in outside the Republican Guard headquarters and other locations throughout Cairo, to protest the "coup" against Mursi.
"The most important thing with the army is that they stay out of politics. We had a legitimate, elected president, and the army came and removed him," he said.
Clashes were repeated across the country.
Thousands of Islamists took to the streets of Alexandria and Assiut to join protests, and in Damanhour, capital of the Beheira province in the Nile Delta, 21 people were wounded in violence between supporters and opponents of Mursi.
Ehab el-Ghoneimy, manager of the Damanhour general hospital, said three people had been wounded with live bullets, others were wounded with birdshot, rocks, or had been hit with rods.
In the Suez city of Ismailia, soldiers fired into the air as Mursi supporters tried to break into the governor's office. The Islamists retreated and there were no casualties, security sources said.
State television and radio also reported clashes in the Nile Delta towns of Gharbeya and Beheira, in Qena south of Cairo and the rural province of Fayoum. No casualties were reported.
Egypt's liberal coalition issued an "urgent call" for its supporters to take to the streets in response to Islamist protests.
In the Sinai Peninsula bordering Israel, two police officers were shot dead on Friday by unknown gunmen in El Arish, medical sources said in an incident not believed linked to the protests.
Overnight, gunmen fired rocket-propelled grenades at army checkpoints guarding an airport there and rocketed a police station near the border with the Palestinian territory of Gaza, killing one soldier and wounding two.
An army spokesman said the army in the Sinai Peninsula was "on alert". He denied an earlier report by state-owned media Al-Ahram that a state of emergency had been imposed in the South Sinai and Suez provinces, which had caused a spike in oil prices from international markets on edge over the unrest.
Egypt's interim head of state, appointed on Thursday, began work to prepare the country for new elections, dissolving parliament by decree. State television also said he appointed a new head of intelligence.
Foreign diplomacy was being handled by the head of Egypt's armed forces on Friday, as General Abdel Fattah al-Sisi, who announced Mursi's overthrow on Wednesday, called Saudi King Abdullah to reassure him Egypt was stable.
(Reporting by Asma Alsharif, Mike Collett-White, Alexander Dziadosz, Seham El-Oraby, Shaimaa Fayed, Maggie Fick, Alastair Macdonald, Shadia Nasralla, Tom Perry, Yasmine Saleh, Paul Taylor, and Patrick Werr in Cairo, Abdelrahman Youssef in Alexandria and Yursi Mohamed in Ismailia; Writing by Mike Collett-White; Editing by Peter Graff)