“The Russian army is setting up checkpoints at the entrances to Kherson,” Kulekhaev said on Facebook. “Kherson was and will remain Ukrainian.”
Russian forces have invaded Ukraine from several axes, including from Crimea, since Russian President Vladimir Putin ordered the offensive.
Videos spread on social media in Kherson showed the Russian army entering the city.
“Today I am responsible for the life of our city, and I offer protection as much as our means allow,” said Kulekhaev.
He asked residents not to leave their homes, explaining: “The most important value this city has is your life. This is not a battle, this is a war.”
Kherson is home to about 280,000 people, located north of the Crimean peninsula, which Moscow annexed in 2014.
Russia had announced that it had surrounded the city on Sunday. The Russian army also reported that it had taken control of the city of Berdyansk, which overlooks the Sea of Azov, northwest of Crimea.