A Russian missile strike wounded 10 people in the central city of Dnipro and three people, including a child, were hurt in an attack on the southern city of Kherson.
Three of the victims in Dnipro were in hospital with moderate wounds, regional governor Serhiy Lysak reported, and a transport facility was damaged.
More than 10 other buildings including a bank, a hotel and an administrative building were damaged, he said.
Photos posted by Lysak on the Telegram messaging app showed a partially destroyed building where people were clearing outbroken glass and a twisted metal structure.
“The photo shows the only thing the ‘Russian world’ is capable of. Destroying everything it touches,” he said.
A seven-year-old girl was injured in Russian shelling of the centre of Kherson, regional governor Oleksandr Prokudin said.
Ukraine’s general prosecutor’s office said two other people were wounded when one of the shells hit a roof of a private house, and posted a photo online showing a partially destroyed building with a gaping hole in one side.
Reuters could not independently verify the reports.
Russia has regularly carried out strikes on Ukrainian centres far from the front line though it denies deliberately targeting civilians.
Thursday was a national holiday commemorating post-Soviet independence. The anniversary fell 18 months after Russia launched its full-scale invasion of its neighbour.
In a pre-recorded address, President Volodymyr Zelensky thanked the Ukrainian people for their “indomitability” and said each individual had a role to play.
“When we celebrate Ukraine’s independence, everyone can feel a part of it,” he said. “And today I want to dedicate these congratulations to you. To you, who is giving Ukraine its independence.”
He said the day was “a holiday of free people” and freedom “is a value for each of us, and we are fighting for it”.
The country’s military intelligence chief Kyrylo Budanov said on Telegram that the fight for independence “continues to this day – now with the imperial aggressor” Russia.
Kyiv’s security service chief Vasyl Maliuk also said on Telegram that the holiday had taken on “new meanings” during the war with Moscow.
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“It became not only the personification of our right to life and freedom, but also a symbol of heroism and bravery,” Maliuk said.
A series of events were planned across Ukraine to mark the day.
In Brussels, EU government buildings were lit up in the blue-and-yellow colours of Ukraine overnight, and Ukrainian flags were hoisted along EU ones.
EU Commission chief Ursula von der Leyen commended Ukrainians for their “courage, their force and enduring hope in a future of peace and prosperity in a united Europe”.
“They are an inspiration to all Europeans,” she said on social media.
Additional reporting by Agence France-Presse