* Protesters dead from bullet wounds, sources say
* Brotherhood declared terrorist organisation last month
* Vote on new constitution in less than two weeks (Updates death toll, adds context)
By Maggie Fick
CAIRO, Jan 3 (Reuters) - Four people were shot dead as supporters of the Muslim Brotherhood clashed with police in Cairo and in cities across Egypt on Friday, defying an ever-widening state crackdown on the Islamist movement.
Islamists opposed to the army's overthrow of President Mohamed Mursi in July have been holding daily demonstrations, even after the army-backed government declared his Brotherhood a terrorist group last week, increasing the penalties for dissent.
The government is using the new classification to detain hundreds of Brotherhood supporters. Thousands more, including top leaders of the group, have been in jail for months, arrested in the aftermath of the army takeover.
A male protester and a woman were shot dead during clashes between pro-Brotherhood demonstrators and police in the coastal city of Alexandria, medical and security sources said. There were conflicting reports as to whether the woman was a protester or an onlooker.
Another demonstrator was shot dead by police in the Suez Canal city of Ismailia after a march set off from a mosque after midday prayers, medical sources said.
In the rural province of Fayoum, southwest of Cairo, a protester died from a bullet wound to the head, local Health Ministry official Medhat Shukri told Reuters.
The Brotherhood - the country's oldest and best organised Islamist movement - is not only under pressure due to mass arrests of its members, orders that the assets of Brotherhood leaders be frozen, and the designation of the group as a terrorist organisation.
A new constitution to be voted on this month also bans religiously based political parties and gives more power to the military.
The referendum on Jan 14 and 15 is a milestone in the roadmap which the army-backed authorities say will pave the way for a return to democratic rule by mid-year.
The charter would be a further step toward the complete removal of the Brotherhood from public life after it won every election in Egypt since veteran autocrat Hosni Mubarak was overthrown in the 2011 uprising.
The movement was banned by a court in September, but the terrorist designation gives the state greater power to carry out arrests of any citizen seen to display loyalty to the group.
Authorities have pledged to secure the referendum, despite the daily street protests and frequent bomb attacks against the security services over the past months.
They generally blame the Brotherhood for the unrest, including a suicide bomb attack for which a radical Sinai-based group called Ansar Bayt al-Maqdis claimed responsibility.
The Brotherhood condemned that attack, which killed 16 people, and says it is committed to peaceful activism.
Around 350 police and soldiers have been killed in bombings and shootings since Mursi was ousted.
A conservative estimate puts the overall death toll since Mursi's fall at well over 1,500. Most of those killed have been Mursi supporters, including hundreds gunned down when the security forces cleared a protest vigil outside a Cairo mosque. (Reporting by Yusri Mohamed in Ismailia and Mohamed Abdalla in Cairo; Writing by Maggie Fick; Editing by Alison Williams)