Eight were confirmed dead, including a child and a baby, after an overnight drone strike on the southern port city of Odesa, a regional official said.
Separate shelling attacks on the frontline regions of Kharkiv in the northeast and Kherson and Zaporizhzhia in the south killed another three people, Ukrainian officials said.
"Russia continues to hit civilians," Zelensky said in a post on social media.
"We need more air defences from our partners. We need to strengthen the Ukrainian air shield to add more protection for our people from Russian terror. More air-defence systems and more missiles for air-defence systems save lives."
Ukraine is currently on the back foot in the two-year war as a crucial $60 billion aid package is held up in the United States Congress.
In Odesa, "a nine-storey building was destroyed as a result of an attack by Russian terrorists", Interior Minister Igor Klymenko said in a Telegram post Saturday.
Around 10 people were still unaccounted for, with almost 100 rescuers set to continue a search and rescue operation overnight.
Footage from the scene showed several floors of a residential building collapsed and its facade ripped off.
Ukraine's air force said falling debris from Russian drones it had shot down fell onto residential buildings in both Odesa and Kharkiv.
- 'Difficult situation' -
The attacks came as Russia sought to press its advantage on the battlefield.
Kyiv has acknowledged that it is heavily outgunned and outnumbered, facing ammunition shortages and aid delays.
Half of all promised Western ammunition arrives late, the defence minister has said.
Russian forces have advanced westwards following last month's capture of the key eastern town of Avdiivka, seizing several small villages in recent days.
Visiting frontline military posts on Saturday, Ukraine's new Commander-in-Chief Oleksandr Syrsky said "the situation at the front remains difficult, but controlled".
Kyiv appeared to have launched its own overnight drone attack, with the Russian defence ministry claiming early on Sunday to have shot down 38 drones over the occupied territory of Crimea.
An apparent Ukrainian drone also damaged a residential building a day earlier in Saint Petersburg, Russia's second-largest city.
Videos on Russian social media showed what appeared to be a drone spiralling downwards into the building, triggering an explosion that blew out windows and caused small fires.
The city's National Guard division said its preliminary assumption was that the damage was caused by a "falling drone".
Ukrainian media reported that the drone was shot down by Russia's air defences while targeting an oil depot around a kilometre from the crash site.
Kyiv has hit several Russian oil facilities in recent months in what it has called fair retribution for Moscow's attacks on Ukraine's power grid.
Russia's Investigative Committee said separately that four of its officials had been wounded when a Ukrainian drone dropped explosives over the Bryansk border region.
Russia also expressed outrage at a leak of confidential German army talks in which officers allegedly discussed missile strikes on the annexed Crimean peninsula.
German Chancellor Olaf Scholz on Saturday promised a full investigation after the head of Russia's state-run RT outlet posted the alleged leaked recording on social media.
A spokeswoman for the German defence ministry confirmed on Saturday that a secret air force conversation had been tapped, but that they could not say for certain whether any changes had been made to the conversation in the leaked audio file.
Russia's foreign ministry demanded a "prompt" explanation from Berlin, while Moscow's top diplomat Sergei Lavrov said it showed Ukraine's European backers were intent on inflicting a "strategic defeat" on Russia on the battlefield.
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