Mykola Lukashuk, head of the Dnipro regional council, said Russian forces had shelled the outskirts of Velyka Kostromka in the Kryvyi Rih area. A village bordering the Zaporizhzhia region was also shelled. Unverified social media posts reported shelling in the northeastern city of Kharkiv.
Reuters could not independently verify the reports from either side on what was happening on the ground.
Ukraine's border guard service posted a video on its Facebook page showing about 16 weary-looking soldiers in battle fatigues, most of them helmeted and holding rifles, singing the national anthem in a small basement room. One was a woman, holding a dog.
The service said the fighters were border guards helping to defend the Azovstal steel works, the last bastion of Ukrainian forces holding out against Russian invaders in the strategic port city of Mariupol, on the Azov Sea.
Reuters could not verify the video.
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Moscow calls the war a "special military operation" to disarm and "denazify" Ukraine, defend Russian-speaking people from persecution and prevent the United States from using the country to threaten Russia.
Ukraine dismisses Putin's claims of persecution and says it is fighting an unprovoked land grab to fully capture Donetsk and Luhansk, which form the Donbas region.
Western officials said Russia had suffered fewer casualties after narrowing the scale of its invasion but that numbers were still "quite high".
Britain's defence ministry said on Saturday: "Shortcomings in Russian tactical coordination remain." Russia had been forced to merge and redeploy depleted and disparate units from failed advances in northeastern Ukraine, it said in a daily bulletin.