Andriy Sadovyi said another eight people were injured, including one seriously. There are fears that more people may be trapped under the rubble.
Regional head Maksym Kozytskyi said a "critical infrastructure facility" was struck in the city. He gave no details.
Russia's military has not commented on the reported attack.
Mr Sadovyi posted a video showing broken windows of the struck building, which appeared to have four floors. Damaged cars and debris were also seen in the footage.
"Many apartments are damaged - more than 50," the mayor said in a short video message.
He added that about 50 cars were also destroyed.
Mr Sadovyi said two people had been pulled from the rubble a short while ago - but rescue teams were continuing their search.
BBC