US-backed forces were locked in fierce fighting as they pressed the battle against the last shred of the Islamic State group's "caliphate" in eastern Syria on Sunday, a war monitor said.The Syrian Democratic Forces (SDF), supported by a US-led coalition, announced a final push to retake the militant pocket near the Iraqi border late on Saturday, after a pause of more than a week to allow civilians to flee.The Syrian Observatory for Human Rights reported heavy clashes between both sides on Sunday morning, as coalition planes and artillery bombarded militant positions."The battle is ongoing. There were heavy clashes this morning, with landmines going off," said Rami Abdel Rahman, the head of the Britain-based war monitor.The SDF launched an offensive to expel IS from the eastern province of Deir Ezzor in September.The Kurdish-led alliance has since whittled down militant-held territory to a patch of just four square kilometres (one square mile) on the eastern banks of the Euphrates.Up to 600 militants could still remain inside, most of them foreigners, according to SDF spokesperson Mustafa Bali.Since fighting intensified in December, more than 37,000 people, mostly wives and children of militant fighters, have fled out into SDF-held desert areas, the Observatory says.That figure includes some 3,200 suspected militants detained by the SDF, according to the monitor, which relies on sources inside Syria for its information.At the height of their rule, the militants imposed their brutal interpretation of Islamic law on a territory spanning parts of Syria and Iraq that was roughly the size of Britain.But separate military offensives in both countries, including by the SDF, have since retaken the vast bulk of the cross-border "caliphate" they declared in 2014.