A boy sizes up the damage after a Russian strike in Kostiantynivka. AFP
The overnight strikes in the Dnipropetrovsk region in central Ukraine also injured 11 people, with five reported to be in a serious condition.
"It was a terrible night, regional governor Valentin Reznichenko wrote on Telegram, urging residents to shelter when they hear air raid sirens.
"I am asking and begging you... Don't let the Russians kill you," he wrote, adding that Russia had fired a total of 80 rockets at the area.
Most of the casualties were in the town of Marganets, just across the Dnieper River from the Zaporizhzhia nuclear power plant, Europe's biggest.
Regional council head Mykola Lukashuk said the strikes had hit a local power line, leaving thousands of people without electricity.
Ukraine and Russia have accused each other of recent shelling around the plant itself.
Ukraine says Russia has stationed hundreds of troops and stored ammunition at the plant.
The tensions have brought back memories of the 1986 Chernobyl nuclear disaster in then Soviet Ukraine, which killed hundreds of people and spread radioactive contamination over much of Europe.
The plant was captured by Russian troops on March 4 after a battle with Ukrainian forces.
The strikes came a day after major blasts at the Saki airfield, a key military base on the Russian-annexed Crimea peninsula.
Dramatic amateur footage on social media appeared to show panicked holidaymakers fleeing a Crimean beach with young children, as ballooning clouds of grey smoke rose over the horizon.
Russia annexed Crimea in 2014 and has used the region as a staging ground for its attacks, but it has rarely been a target for Ukrainian forces.
Moscow insisted that the explosions were caused by detonating ammunition rather than Ukrainian fire.
Ukraine's army, which for months pleaded for long-range artillery from Western allies, has been hitting targets deeper in Russian-held territory since some started arriving in recent weeks.
Kyiv has also taken credit for several acts of sabotage inside Russian-held territory.
New buyer for Ukraine grain?
The war has severely hampered grain supply from Ukraine, leading to an international food crisis as it is one of the world's biggest producers.
But some ships have been able to leave Ukrainian ports in recent days after a deal with Russia brokered by the United Nations and Turkey.
The first grain shipment to leave on the Sierra Leone-flagged vessel Razoni departed the Ukrainian port of Odessa on August 1 and had been expected to dock in the Lebanese port of Tripoli at the weekend.
But the Ukrainian embassy said a new buyer for the shipment was being sought after the original Lebanese buyer cancelled the order.
A five-month delay after Russia's invasion "prompted the buyer and the shipping agent to reach agreement on the cancellation of the order," the Ukraine embassy said in a statement late Tuesday.
It is currently anchored off the Turkish port of Mersin.