A video posted earlier on Tuesday by Lviv authorities showed a fire burning through the roof of a residential building near the Bernardine monastery.
Separate footage posted on social media showed a drone flying lower over the city and hitting the residential building.
Lviv regional head Maksym Kozytskyi said 32 people were injured in the Russian attack.
In the city of Ivano-Frankivsk, two people were killed and another four – including a six-year-old child – were injured, local officials said.
Various buildings – including a maternity hospital – were damaged in the regional capital.
Ternopil – another western Ukrainian city – was also targeted on Tuesday. A number of direct hits were reported by regional authorities, but no casualties.
In the central Ukrainian city of Vinnytsia, one person was killed and 13 injured, regional head Nataliya Zabolotna said.
In the overnight Russian attacks, five people were killed when Russia targeted 11 Ukrainian regions.
Ukraine’s Air Force said it had managed to shoot down most of the Russian drones and missiles – but admitted that there were multiple direct hits across the country.
Yurii Ihnat, spokesman of the Ukrainian Air Force, said late on Tuesday that a “large number of drones” had entered Ukrainian airspace from the north of the country, “effectively moving in columns”.
“The geography of the strikes during the daytime was broader than at night… It can be said this was one of the largest attacks within a 24-hour period,” he said.
While more than four years of war have left virtually no corner of Ukraine untouched, the west of the country has been hit comparatively less intensely and frequently than other areas nearer the Russian border in the east.
Russia launched a full-scale invasion of Ukraine in February 2022, and the war continues unabated, with Moscow launching near-daily attacks on cities across the country.
Also on Tuesday, the governor of Russia’s western Kursk region said one man was killed and 13 people were injured in a Ukrainian drone attack on an agricultural enterprise.
Talks brokered by the US and aimed at reaching at a peace settlement have stalled since the start of the US-Israeli war with Iran in late February.
“Amid the news the world is drowning in every day, we will not let Ukrainian grief get lost, become just another statistic, a headline that will be casually skipped over,” wrote Ukraine’s First Lady Olena Zelenska on social media.