Volodymyr Zelensky has warned Russian troops to “flee for their lives” as Ukrainian forces attempt to reclaim territory in the southern region of Kherson.
The Ukrainian president said his country’s military has taken back land during the offensive, while Russia claims the assault has failed.
Mr Zelensky vowed Ukrainian troops would chase the Russian army “to the border”, during his nightly address late on Monday.
“If they want to survive - it’s time for the Russian military to run away,” he said. “Go home. Ukraine is taking back its own,” Zelensky said.
Ukraine’s southern command said its troops had launched offensive actions in several directions, including in the Kherson region north of the Crimean peninsula that Russia annexed from Ukraine in 2014.
Oleksiy Arestovych, a senior adviser to Mr Zelensky, said Russian defences had been “broken through in a few hours”.
Ukrainian forces were shelling ferries that Russia was using to supply a pocket of territory on the west bank of the Dnipro river in the Kherson region, he added.
On Tuesday, Ukraine’s Suspilne public broadcaster reported explosions in the Kherson area and city residents reported in social media posts gunfire and explosions, but said it was not clear who was firing.
Ukraine’s military general staff, in an early update, reported clashes in various parts of the country but gave no information on the Kherson offensive.
Meanwhile Russia’s defence ministry said Ukrainian troops had attempted an offensive in the Mykolaiv and Kherson regions but sustained significant casualties, RIA news agency reported.
The “enemy’s offensive attempt failed miserably”, it said.
But a Ukrainian barrage of rockets left the Russian-occupied town of Nova Kakhovka without water or power, officials at the Russian-appointed authority told RIA news agency.
News agencey Reuters could not verify the battlefield reports.
Russian shelling of the port city of Mykolaiv, which has remained in Ukrainian hands despite repeated Russian bombardments, killed at least two people, wounded some 24 and wiped out homes, city officials and witnesses said on Monday.
Ukraine had struck more than 10 sites in the past week and “unquestionably weakened the enemy”, according to a spokeswoman who declined to give details of the offensive, saying Russian forces in the south remained “quite powerful”.