Ukraine's President Volodymyr Zelensky has pledged to retake more areas in the country's eastern Donbas region from Russian forces.
"Throughout this week, more Ukrainian flags have been raised in the Donbas. There will be even more in a week," he said in his evening address.
He spoke after Kyiv said its forces had begun moving into the key eastern town of Lyman and the defence ministry posted a video of soldiers holding up a yellow and blue Ukrainian flag there.
Russia's defence ministry said it had "withdrawn" troops from the town "to more favourable lines".
Mr Zelensky told Russians they would be "killed one by one" as long as President Vladimir Putin, who ordered the invasion of Ukraine in February, remained in power.
"Until you all solve the problem with the one who started it all, who started this senseless war against Ukraine, you will be killed one by one," he said, calling the war "a historic mistake for Russia".
Russia said earlier that its troops had abandoned the key town of Lyman in occupied eastern Ukraine - a stinging defeat that prompted a close ally of Mr Putin to call for the possible use of low-grade nuclear weapons.
The announcement came just a day after Putin proclaimed the annexation of four Ukrainian regions - including Donetsk, where the city of Lyman is located - and placed them under Russia's nuclear umbrella, at a ceremony that was condemned by Kyiv and the West as an illegitimate farce.
"In connection with the creation of a threat of encirclement, allied troops were withdrawn from the settlement of Krasny Liman to more advantageous lines," Russia's defence ministry said, using the Russian name of the town.
President Volodymyr Zelensky later said in a video address that although the Ukrainian flag was flying in the city, "fighting is still going on there".
He also indicated Ukrainian troops had taken the village of Torske, on the main road out of Lyman to the east.
The Russian statement ended hours of official silence after Ukraine first said it had surrounded thousands of Russian troops in the area and then that its forces were inside the city.
Ukraine's defence ministry wrote on Twitter that "almost all" the Russian troops in Lyman had either been captured or killed.
Russia has used Lyman as a logistics and transport hub for its operations in the north of the Donetsk region. Its capture would be Ukraine's biggest battlefield gain since a lightning counteroffensive in the northeastern Kharkiv region last month.
Mr Zelensky said that in the days to come, Ukrainian forces would liberate more towns.
The recent Ukrainian successes have infuriated Putin allies such as Ramzan Kadyrov, the leader of Russia's southern Chechnya region, who said he felt compelled to speak out.
"In my personal opinion, more drastic measures should be taken, right up to the declaration of martial law in the border areas and the use of low-yield nuclear weapons," Mr Kadyrov wrote on Telegram before Mr Zelensky spoke.
Other top Putin allies, including former president Dmitry Medvedev, have suggested Russia may need to resort to nuclear weapons, but Kadyrov's call was the most urgent and explicit.
Mr Putin said last week that he was not bluffing when he said he was prepared to defend Russia's "territorial integrity" with all available means, and on Friday made clear this extended to the new regions claimed by Moscow.
Washington says it would respond decisively to any use of nuclear weapons and has spelled out to Moscow the "catastrophic consequences" it would face.
'Already in Lyman'
The Russian defence ministry's statement made no mention of its troops being encircled at Lyman.
"The Russian grouping in the area of Lyman is surrounded," Serhii Cherevatyi, spokesperson for Ukraine's eastern forces, said hours earlier.
He said Russia had 5,000 to 5,500 troops at Lyman but the number encircled could be lower. He later confirmed Ukrainian troops were inside the town.
Two Ukrainian soldiers taped the yellow-and-blue national flag to the 'Lyman' welcome sign at an entrance to the city, a video posted by the president's chief of staff showed.
"1 October. We're unfurling our state flag and establishing it on our land. Lyman will be Ukraine," one of the soldiers said.
Neither side's battlefield assertions could be independently verified.
Mr Kadyrov said Colonel-General Alexander Lapin, the commander overseeing Lyman, was a "mediocrity" who should be stripped of his medals and sent to the front line. Mr Kadyrov said he had warned army chief General Valery Gerasimov of a looming disaster.
"The general assured me he had no doubts about Lapin's talent for leadership and did not think a retreat was possible," he said.
Ukraine says its capture would allow Kyiv to advance into the Luhansk region, whose full capture Moscow announced in early July after weeks of grinding advances.
"Lyman is important because it is the next step towards the liberation of the Ukrainian Donbas. It is an opportunity to go further to Kreminna and Sievierodonetsk, and it is psychologically very important," Mr Cherevatyi said.
The Donetsk and Luhansk regions make up the wider Donbas region that has been a major focus for Russia since soon after the start of Moscow's invasion on 24 February in what it called a "special military operation" to demilitarise its neighbour.
Mr Putin proclaimed the Donbas regions of Donetsk and Luhansk and the southern regions of Kherson and Zaporizhzhia to be Russian land on Friday - a swathe of territory equal to about 18% of Ukraine's total surface land area.
Ukraine and its Western allies branded Russia's move as illegal. Kyiv vowed to continue liberating its land of Russian forces and said it would not hold peace talks with Moscow while Putin remained president.
Mykhailo Podolyak, an advisor to Mr Zelensky, mocked the Kremlin ceremony on Friday announcing the annexation.
"Now Russian troops are leaving another strategic city and propagandists are looking for culprits. Reality can hurt if you live in fantasy world," Mr Podolyak wrote on Twitter.
Retired US General Ben Hodges said a Russian defeat in Lyman would be a further blow to the morale of Moscow's troops and a major political and military embarrassment for Putin.
"This puts in bright lights that his claim is illegitimate and cannot be enforced," he said.