"Unfortunately, there are already six dead. The rescue operation continues," Oleksandr Vilkul said on the Telegram messaging app.
Ukrainian officials had said earlier on Tuesday that three people had been killed and many more were trapped under the rubble of a five-storey apartment building.
Ukraine's President Volodymyr Zelenskiy who was born in Kryvyi Rih, also condemned the attack on his hometown.
"Russian killers continue their war against residential buildings, ordinary cities and people," he said on Telegram messaging app.
During the early hours of Tuesday, air raid sirens blared across the whole of Ukraine, with Kyiv's military officials saying air defence forces destroyed all Russian missiles targeting the Ukrainian capital.
Ukraine's top military command said that air forces destroyed 10 out of 14 cruise missiles Russia launched on Ukraine and one of the four Iranian-made drones.
It was not immediately clear how many missiles hit Kryvyi Rih and where the Russia-launched drones struck their targets.
There was no immediate comment from Russia about the reported strikes. Both Russia and Ukraine deny targeting civilians in the war which Russia launched on its neighbour nearly 16 months ago.
The mayor of the city of Kharkiv in Ukraine's east said on his Telegram channel that Russian drones hit civilian infrastructure there, striking a warehouse and a utility firm's building. There was no immediate information about casualties.
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