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Turkish President Tayyip Erdogan vowed to press on with rescue and recovery efforts more than a week after a powerful quake ripped through his country and neighbouring Syria, with an elderly woman the latest to be pulled from the rubble.

The combined death toll in Turkey and Syria has climbed over 41,000, and many survivors are enduring near-freezing winter temperatures, having been left homeless by the devastation in cities in both countries.

"We will continue our work until we remove the last citizen left under the collapsed buildings," Erdogan said late on Tuesday after a cabinet meeting held at the headquarters of the Disaster and Emergency Management Authority (AFAD).

Damage assessment of buildings, of which tens of thousands were destroyed, will be completed in a week and reconstruction will begin within months, he said.

"We will rebuild all the houses and workplaces, destroyed or made uninhabitable by the earthquake, and hand them over to the rightful owners," he added. More than 105,000 people were injured in the quake, he said, with more than 13,000 still being treated in hospital.

Overnight, a 77-year-old woman named Fatma Gungor was pulled alive from the rubble of a seven-storey apartment block in the city of Adiyaman, some 212 hours after the first earthquake, media reports said.