The attack caused a huge fire and sent dark smoke billowing into the sky, footage circulated by President Volodymyr Zelenskiy showed.
A Reuters reporter saw the charred husk of a shopping complex with a caved-in roof.
Firefighters and soldiers were pulling out mangled pieces of metal as they searched for survivors.
"We don't understand how many people could be remaining under the rubble," the regional rescue service chief said on television.
Zelenskiy said more than 1000 people were in the shopping centre at the time of the attack.
He gave no details of casualties but said: "It is impossible to even imagine the number of victims."
"It's useless to hope for decency and humanity from Russia," Zelenskiy wrote on the Telegram messaging app.
A rescue operation was underway and nine of the wounded were in a serious condition, Kyrylo Tymoshenko, deputy head of the presidential office, said.
"Rescuers from all surrounding areas are heading to the scene to put out the fire and work to liquidate the consequences," Tymoshenko said.
Kremenchuk, an industrial city of 217,000 before Russia's February 24 invasion of Ukraine, lies on the Dnipro river in the region of Poltava and is the site of Ukraine's biggest oil refinery.
There was no immediate comment from Russia, which denies deliberately targeting civilians.
"We need more weapons to protect our people, we need missile defences," Andriy Yermak, head of the president's office, said.
Vadym Denysenko, an interior ministry adviser, said Russia could have had three motives for the attack.
"The first, undoubtedly, is to sow panic, the second is to... destroy our infrastructure, and the third is to... raise the stakes to get the civilised west to sit down again at the table for talks," he said.
Russia, which captured Ukraine's eastern frontline city of Sievierodonetsk over the weekend after a weeks-long assault, has stepped up missile strikes on targets across Ukraine in recent days.
Missiles slammed into an apartment block and landed close to a kindergarten in the Ukrainian capital on Sunday, killing one person and wounding several more people.