The UN refugee agency says one million people have fled Ukraine since Russia's invasion less than a week ago, an exodus without precedent in this century for its speed.
The tally from UNHCR amounts to more than 2 percent of Ukraine's population on the move in under a week. The World Bank counted the population at 44 million at the end of 2020.
The UN agency has predicted that up to 4 million people could eventually leave Ukraine but cautioned that even that projection could be revised upward.
In an email, UNHCR spokesperson Joung-ah Ghedini-Williams wrote: "Our data indicates we passed the 1M mark" as of midnight in central Europe, based on counts collected by national authorities.
On Twitter, the UN High Commissioner for Refugees, Filippo Grandi, wrote: "In just seven days we have witnessed the exodus of one million refugees from Ukraine to neighbouring countries."
Syria, whose civil war erupted in 2011, currently remains the country with the largest refugee outflows - at more than 5.6 million people, according to UNHCR figures. But even at the swiftest rate of flight by refugees out of Syria, in early 2013, it took at least three months for one million refugees to leave that country.
UNHCR spokesperson Shabia Mantoo said on Wednesday that "at this rate" the outflows from Ukraine could make it the source of "the biggest refugee crisis this century."
Battle for Kherson
Ukrainian President Volodymyr Zelenskyy's office says fighting is still occurring around the port city of Kherson, which Russian officials have said is in their "complete control."
Mr Zelenskyy's office told The Associated Press that it could not comment on the situation there while the battle was still being waged.
But the mayor of Kherson, Igor Kolykhaev, said Russian soldiers were in the city and came to the city administration building. He said he asked them not to shoot civilians and to allow them to gather up the bodies from the streets.
"I simply asked them not to shoot at people," he Kolykhaev said in a statement. "We don't have any Ukrainian forces in the city, only civilians and people here who want to LIVE."
Kherson, a city of 300,000 people, is strategically located on the banks of the Dnieper River near where it flows into the Black Sea. If Russian troops take the city, they could unblock a water canal and restore water supplies to the Crimean Peninsula.
Ukraine cut off the fresh water supply to Crimea by damming a canal that had supplied 85 per cent of the peninsula's needs before Russia annexed Crimea in 2014.
The battle in the Kherson region began last Thursday, the first day of the invasion, and by the next day the Russian forces were able to take a bridge that connects the city with territory on the western bank.
'We'll drive them out'
In a video address to the nation early on Thursday, Ukrainian President Volodymyr Zelenskyy gave an upbeat assessment of the war and called on Ukrainians to keep up the resistance.
"We are a people who in a week have destroyed the plans of the enemy," he said.
"They will have no peace here. They will have no food. They will have here not one quiet moment."
Mr Zelenskyy didn't comment on whether the Russians have seized several cities, including Kherson.

Video obtained by Reuters shows a convoy of military trucks and tanks in a street in Kherson. Credit: Reuters
"If they went somewhere, then only temporarily. We'll drive them out," he said.
He said the fighting is taking a toll on the morale of Russian soldiers, who "go into grocery stores and try to find something to eat."
"These are not warriors of a superpower," he said. "These are confused children who have been used."
He said the Russian death toll has reached about 9,000.
"Ukraine doesn't want to be covered in bodies of soldiers," he said. "Go home."
Russia releases figure for Russian troop deaths
Russia says that 498 of its troops have been killed in Ukraine, its first announced death toll since President Vladimir Putin launched his invasion.
Major General Igor Konashenkov on Wednesday rejected reports about "incalculable losses" of the Russians as "disinformation."
The UN said on Wednesday it had recorded 227 civilian deaths in Ukraine since Russia's full-scale invasion began, warning that the true toll was likely far higher.
"OHCHR believes that real figures are considerably higher," it said in a statement, adding that fighting had made it hard to corroborate exact figures.
The UN rights agency said that most of the casualties were "caused by the use of explosive weapons with a wide impact area, including shelling from heavy artillery and multi-launch rocket systems, and air strikes."
It pointed out that its tally, though an underestimate, was "more than the total number of civilian casualties recorded by OHCHR in the conflict zone of eastern Ukraine" from 2012 to 2018.
The toll is also more than double the 102 deaths reported by UN rights chief Michelle Bachelet on Monday.
Additional reporting by AFP, Reuters.