The unit's claim on Wednesday appeared to back up comments by Yevgeny Prigozhin, the head of the Wagner private army, who the previous day said the Russian brigade had abandoned its positions.
Colonel General Oleksandr Syrskyi, who heads Ukraine's ground forces, said Russian units in some parts of Bakhmut had retreated by up to 2km as the result of counter attacks. He gave no details.
Reuters was unable to independently confirm the situation on the ground. Wagner units have led a months-long Russian assault on the eastern city, suffering heavy losses, but Ukrainian forces say the offensive is stalling.
"The special military operation continues. This is a very difficult operation, and, of course, certain goals have been achieved in a year," Tass new agency quoted Kremlin spokesman Dmitry Peskov as telling a Bosnian Serb television channel.
"We managed to beat up the Ukrainian military machine quite a bit," said Peskov, citing Russian missile strikes in Ukraine. "This work will continue".
His comments did not address claims that Russia's 72nd Separate Motor-rifle Brigade had abandoned positions on the southwestern outskirts of Bakhmut.
In a statement, Ukraine's Third Separate Assault Brigade said: "It's official. Prigozhin's report about the flight of Russia's 72nd Independent Motorized Rifle Brigade from near Bakhmut and the '500 corpses' of Russians left behind is true."
A Russian brigade is typically formed of several thousand troops. Bakhmut is the primary target of Moscow's huge winter offensive and scene of the bloodiest ground combat in Europe since World War Two.
"Our army is fleeing. The 72nd Brigade pissed away three square km this morning, where I had lost around 500 men," Prigozhin said on Tuesday, complaining his troops were receiving only 10% of the shells they needed.
Prigozhin has clashed repeatedly with Russia's defence ministry and expressed concerns about a long-promised Ukrainian counter-offensive to recapture some territory Russia occupied after the 2022 invasion.
Ukrainian Deputy Defence Minister Hanna Malyar wrote on Telegram that pro-Kyiv units had not lost a single position in Bakhmut on Wednesday.
'SITUATION REMAINS DIFFICULT'
Russian troops invaded Ukraine in February 2022 in what Moscow calls a special military operation and initially captured significant amounts of territory. But Kyiv's forces pushed back and are planning another counteroffensive. Western officials estimate more than 200,000 Russian soldiers have been killed or wounded
In his evening video address, Ukrainian President Volodymyr Zelenskiy named the Third Brigade and noted its report "about the flight of Russia's 72nd Independent Motorized Rifle Brigade from near Bakhmut".
In Brussels, NATO's top military official said the war would increasingly be a battle between large numbers of poorly trained Russian troops with outdated equipment and a smaller Ukrainian force with better Western weapons and training.
Admiral Rob Bauer, a Dutch military officer who is chair of NATO's military committee, noted Russia was now deploying significant numbers of T-54 tanks - an old model designed in the years after World War 2.
In Washington, US Attorney General Merrick Garland said he had authorized the first transfer of forfeited Russian assets for use in Ukraine.
In Warsaw, Poland summoned Russia's ambassador over an incident involving a Russian fighter jet and a Polish border guard aircraft over the Black Sea, a Polish foreign ministry spokesman said.
