Scores of Ukrainian soldiers are feared to have been killed after an attack on a military barracks on Friday.
About 200 soldiers were sleeping in the barracks when three Russian missiles hit the base on the northern edge of Mykolaiv, a source told the BBC. Some 57 injured people were being treated in hospitals, another source said.
There are no official casualty figures.
Rescue workers crawling over giant piles of rubble found a survivor on Saturday - 30 hours after the attack.
The man was carried down the steep mountain of rubble in a stretcher and taken away by ambulance towards the city centre.
However, the temperature in Mykolaiv last night was minus 6C and it's feared there may not be many more survivors.
Vitaly Kim, head of the regional administration, blamed Russians for hitting "sleeping soldiers with a rocket in a cowardly manner".
"At least 50 bodies have been recovered, but we do not know how many others are in the rubble," a soldier named Maxim was quoted as saying by AFP news agency.
And the rescue work at the site is being carried out amid fears of new Russian attacks.
With a crane pulling away chunks of concrete and twisted metal, suddenly soldiers guarding the base shouted at journalists to run for cover, as the sound of a plane, or possibly a missile, could be heard overhead.
Seconds later a dull explosion could be heard somewhere to the north-east.
Near the army base is the site of an attack which killed nine civilians queueing outside a shop. There we meet Yulia, who is in tears as she tells the BBC what happened.
"I can't tell you how scared we are," she says. "We all live in the cellar now. When the bombs hit the army base - everything shook."
Two blocks away, Andre Ansimov is trying to clear up from another Russian strike. The crater is so big, a car and a house fell into it.
Only by chance were Andre and his family away when the bomb hit.
"This isn't a war between armies," he said. "The Russians are trying to destroy us all."
Mykolaiv has been blocking Russia's advance west along the Black Sea coast for weeks. Overnight there was more heavy shelling in the south of the city.
There are also reports that Russian reconnaissance teams, sent in to locate military targets for attack, have been found there.