The regional governor's adviser, Natalia Babachenko, said 30 people were confirmed dead so far and more than 30 were in hospital, including 12 in a serious condition.
Between 30 to 40 people could still be trapped under debris, she said.
Emergency workers said they had heard people screaming for help from underneath piles of debris from the nine-storey apartment block in the east-central city and were using moments of silence to help direct their efforts.
Freezing temperatures added to rescuers' concerns.
A group of firefighters found a lightly-dressed woman still alive more than 18 hours after the attack.
They carried her to safety in their arms.
"The chances of saving people now are minimal," Dnipro's Mayor Borys Filatov told Reuters.
"I think the number of dead will be in the dozens."
Ukraine's Air Force said the apartment block was struck by a Russian Kh-22 missile, which is known to be inaccurate and that Ukraine lacks the air defences to shoot down.
The Soviet-era missile was developed during the Cold War to destroy warships.
Filatov said two stairwells including dozens of flats were destroyed.
Russia fired two waves of missiles at Ukraine on Saturday, striking targets across the country as fighting raged on the battlefield in the eastern towns of Soledar and Bakhmut.
In a statement on Sunday about its previous day of strikes, the Russian defence ministry did not mention Dnipro as a specific target.
"All assigned objects were hit. The targets of the strike have been achieved," it said.
The United Kingdom followed France and Poland on Saturday with promises of further weapons for Ukraine, saying it would send 14 of its Challenger 2 main battle tanks as well as other advanced artillery support in the coming weeks.
In Ukraine's eastern Donbas region - the focal point of Russia's drive to capture more territory - Ukraine's forces were battling around the small salt-mining town of Soledar.
Serhiy Cherevatyi, spokesman for Ukraine's eastern command, told Ukrainian television that Russian forces had shelled the area around Soledar and Bakhmut 234 times in the past 24 hours.
Russia said on Friday that its forces had taken control of Soledar, which had a pre-war population of 10,000, in what would be a minor advance but one that would have psychological importance for Russian forces after months of battlefield setbacks.
Ukraine insisted on Saturday that its forces were battling to hold the town but officials acknowledged the situation was difficult, with street fighting raging and Russian forces advancing from various directions.
The US-based Institute for the Study of War said it was highly unlikely that Ukrainian forces still held positions within Soledar itself.
Reuters could not immediately verify the situation in the town.
An ammunition explosion caused by "careless" handling of a grenade in Russia's Belgorod region on the border with Ukraine killed three soldiers and injured 16, Russian news agencies reported.
The blast occurred in a cultural centre repurposed for Russia's armed forces to store ammunition, state news agencies reported, citing local emergency services for the toll.
Another eight service personnel were still reported missing as of Sunday evening, the Interfax news agency reported.
"As a result of the unintentional detonation of a hand grenade by a sergeant in a dormitory... a fire broke out. Sixteen servicemen, including the culprit, have been taken to hospital. Three more died," local emergency services said in a statement cited by news agencies.
The 112 and Baza Telegram channels said the dead and injured were conscripts called up to fight in Ukraine under a mobilisation drive.
