BAGHDAD (Reuters) - U.S.-backed Iraqi forces on Friday captured one of the four districts making up the Islamic State-held enclave in Mosul, a military statement said.

The elite Counter Terrorism Service (CTS) units took the Sihha district, about one week after the start of operations to dislodge the militants from the enclave and finish off the offensive to capture Mosul, now in its eighth month.

With the loss of Sihha, the enclave has shrunk to three districts alongside the western bank of the Tigris river -- the densely populated Old City center, Zanjili and the Medical City.

Iraqi government forces retook eastern Mosul in January and began a new push on May 27 to capture the enclave where about 200,000 people are trapped in harrowing conditions.

The Mosul offensive started in October with air and ground support from a U.S.-led international coalition. It has taken much longer than expected as Islamic State is fighting in the middle of civilians, slowing the advance of the assailants.

At least seven civilians were killed and 23 wounded by Islamic State mortar shells as they tried to flee Zanjili on Thursday, Iraqi police said.

The fall of Mosul would, in effect, mark the end of the Iraqi half of the ‘’caliphate’’ declared in 2014 over parts of Iraq and Syria by Islamic State leader Abu Bakr al-Baghdadi, in a speech from a historic mosque in the old city.

In Syria, Kurdish forces backed by U.S.-air strikes are besieging Islamic State forces in the city of Raqqa, the militants’ de facto capital in that country.

About 700,000 people, about a third of the pre-war population of Mosul, have already fled, seeking refuge either with friends and relatives or in camps.