The police chief of the war-torn region of Donetsk said the long-range Grad rockets hit a market in an eastern district facing roads that have come under attack from separatist militias in recent days.
At least 10 people died Saturday when heavy rockets fell on residential districts of Ukraine's strategic government-held port of Mariupol in the wake of Russian-backed rebels' rejection of peace talks.
The southeastern Sea of Azov city of nearly 500,000 sits on a highway connecting guerrilla-held regions to the east and the Black Sea peninsula of Crimea that Russia annexed from Ukraine in March.
A massive rebel assault on the port in August led to intense fighting that saw Kiev repel the attack at a heavy cost that soon prompted Ukrainian President Petro Poroshenko to agree to a September 5 truce.
That deal was followed by still more clashes that killed at least 1,500 people and was ultimately rejected by the rebels on Friday.
The separatist leader of Donetsk vowed on Friday to escalate the nine-month campaign and seize lands in southeastern Ukraine that are currently under control of the pro-Western authorities.
The pro-Kiev volunteer Azov regiment based around the city said there were "many wounded" in the attack. The Mariupol administration said the rockets hit a large market in one of the city's main residential districts.
"Right now, there are problems with the cell phone network so it is impossible to call relatives who live in that part of town," Mariupol resident Eduard told AFP by telephone.
"Obviously, everyone in the city is very scared. The rebels have already seized the airport. And now they are starting to destroy Mariupol itself."
There was no immediate comment on the battle from rebel leaders.