The city of Severodonetsk in eastern Ukraine has been under bombardment by Russian forces for weeks, causing people to take refuge in cellars and basements to escape the fighting.
Police on Wednesday searched schools and other buildings in the hope of finding those sheltering to offer them help and in some cases transport to other cities such as Lviv.
Despite living in desperate conditions, with little food, the majority refuses to leave, British broadcaster Sky News reported.
Some residents said they were distrustful of the police while other did not want to be left destitute in another city they do not know.
Ukrainian authorities urged people living in the Donbas to evacuate now, ahead of an impending Russian offensive, while there is still time.