Russian troops have been told to flee Ukraine by president Volodymyr Zelensky as his forces launched a major offensive.
Ukraine's military is attempting reclaim invaded territory near the southern city of Kherson.
Russia responded to the attack by claiming the assault had failed, although it is thought to have seriously disrupted their supply chain.
The ongoing war has claimed thousands of lives and displaced millions but recent weeks have seen intense fighting result in something of a stalemate, with Russia struggling to make further gains.
There have been rumours of Russian troops deliberately injuring themselves in order to avoid battle, while 1,000 refused to return from duties in Kazakhstan.
Speaking in his daily address on Monday, Zelensky said Ukrainian troops would pursue Russia's soldiers “to the border” and advised them to return home if they wanted to survive the conflict.
"If they want to survive - it's time for the Russian military to run away. Go home," he said.
"Ukraine is taking back its own.”
Oleksiy Arestovych, a senior adviser to Zelensky, said Russian defences had been "broken through in a few hours".
He added that Ukraine was shelling ferries used by Russia to supply territory they currently hold on the west bank of the Dnipro river.
Russia's defence ministry claimed, via the RIA news agency, that Ukrainian troops had attempted an offensive in the Mykolaiv and Kherson regions but had suffered significant casualties as a result.
The "enemy's offensive attempt failed miserably", it said.
Nova Kakhova, currently occupied by Russia, was left without water or power after being hit by Ukrainian rockets, officials told RIA.
At least two people were killed by Russian shells in the Ukrainian-held port town of Mykolaiv.
A mission from UN nuclear watchdog the International Atomic Energy Agency to the Russian-occupied Zaporizhzhia nuclear plant was due to arrive in Kyiv on Monday.
Both countries have accused each other of shelling the plant, Europe’s largest, and threatening to spark a disaster worse than Chernobyl.
Russian proxy officials claim that they will ensure inspectors’ safety.
