Russian missiles have slammed into the central Ukrainian city of Kryvyi Rih, killing at least four people, including a 10-year-old girl and her mother, officials say.
Several other people were buried under rubble.
A video posted by President Volodymyr Zelenskiy showed smoke billowing from a gaping hole smashed in the side of a nine-storey residential building, and another four-storey building almost levelled.
"Tragic news. Four people have already died in Kryvyi Rih," Serhiy Lysak, the regional governor, wrote on the Telegram messenger app. Mayor Oleksandr Vilkul said they included a 10-year-old girl and her 45-year-old mother.
Zelenskiy, who grew up in the steel-producing city with a pre-war population of more than 600,000, said the strikes had hit a residential building and a university building.
"This terror will not frighten us or break us. We are working and saving our people," he said on the Telegram app.
More than 200 rescuers were trying to save people trapped under the rubble. Vilkul said there could be as many as eight.
Emergency services said at least 43 people had been wounded. Air Force spokesman Yuriy Ihnat said the attack appeared to have been carried out with ballistic missiles.
Kherson, now a frontline city in southern Ukraine after being liberated from Russian forces in November, was struck at least twice.
An early morning rocket attack killed a 60-year-old utility worker and wounded four others as they were out on the street doing their jobs, the regional military administration said.
A 65-year-old man driving his car was badly wounded in the second strike and died on the way to hospital, regional governor Oleksandr Prokudin said on Telegram.
A Ukrainian artillery strike on the partially occupied province of Donetsk killed two people and wounded six in the regional capital, according to Denis Pushilin, the Moscow-installed leader of the illegally annexed province.
A bus was also hit as Ukrainian forces shelled the city of Donetsk multiple times on Monday, Mr Pushilin said.
Neither side's claims could be independently verified.
Russian Defence Minister Sergei Shoigu said Ukraine's counteroffensive was not going as planned and it was wasting billions of dollars of weapons supplied to it by the West.
He told a military conference on Monday that Ukraine was "desperately hurling new forces" into attacks on Russian positions, but had failed to advance.
Kremlin spokesman Dmitry Peskov told reporters "the multibillion dollar resources that were transferred by NATO countries to the Kyiv regime are actually being spent inefficiently and aimlessly".
He said this was why Kyiv was resorting to "acts of desperation" such as a drone attack against Moscow's business district on Sunday.
The strike damaged two office buildings a few miles from the Kremlin.
With Reuters
