An art school in Mariupol, where about 400 people were sheltering, was attacked, the city council says.
Officials say the building has been destroyed and people could be trapped under the rubble.
Ukraine's president says Russia's attacks on the southern port city of Mariupol will be remembered for centuries.
Volodymyr Zelensky alleges that Russia committed war crimes in the city, where heavy fighting has now reached the centre.
It comes as officials there allege thousands of residents have been forcibly taken to Russian cities. The BBC is yet to verify the claim.
The UK defence ministry, meanwhile, says Russia had failed to gain control of the air and is relying on weapons launched from its own airspace.
Ukraine's military says Russia paused offensive operations on Saturday, instead replenishing supplies and repairing equipment.
More than 3.3 million people have fled the conflict, which has been raging for more than three weeks.
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