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More than a dozen countries have urged their citizens to leave Ukraine amid warnings from Western powers that an invasion by Russia could be imminent.
The US, Germany and UK are among those who told foreign nationals to leave.
Ukraine's President Volodymyr Zelensky said warnings of imminent invasion could stoke panic, which he said was "the best friend of our enemies".
Moscow has amassed an estimated 100,000 troops along Ukraine's border but denies any intent to invade.
The White House has warned that an invasion could happen at any time, and that an invasion could begin with bombing from the air. Russia characterised such allegations as "provocative speculation".
President Zelensky of Ukraine said that if Western powers had any firm evidence of an impending invasion, he had yet to see it.
"I think there is too much information in the media about a deep, full-scale war," he said.
"We understand all the risks, we understand that they exist. If you or someone else has additional 100% reliable information about the Russian Federation's invasion of Ukraine... please share it with us."
Diplomatic efforts continued on Saturday. French President Emmanuel Macron spoke to Russian leader Vladimir Putin for more than 90 minutes by phone, telling him that "a sincere dialogue was not compatible with escalation", according to notes released by the French embassy.
US President Joe Biden spoke to the Russian president in turn for just over an hour. US Secretary of State Antony Blinken had earlier told his Russian counterpart, Mr Lavrov, that diplomatic channels remained open.
Many countries, including Australia, Italy, Israel, the Netherlands and Japan have told their citizens to leave Ukraine. Some have evacuated diplomatic staff. On Saturday, Slovakia became the latest country to withdraw family members of embassy staff from the country.
Non-essential staff have been ordered to leave the US Embassy in Ukraine's capital Kyiv, and consular services will be suspended from Sunday, although "a small consular presence" will remain in the western city of Lviv "to handle emergencies".
Canada is also moving its embassy staff to Lviv, near the border with Poland, Canadian media reported. UK ambassador to Ukraine Melinda Simmons has tweeted that she and a "core team" are staying in Kyiv.
The US has also pulled some 150 troops who were training Ukrainian soldiers out of the country, citing an abundance of caution. And Dutch airline KLM announced it would stop flying to Ukraine, effective immediately, Dutch media said.
Russia itself is also making changes, saying it would "optimise the staffing" of its diplomats in Ukraine, citing "possible acts of provocation by the Kyiv regime or third countries".
With foreign embassies withdrawing staff and a host of countries now telling their citizens to leave Ukraine, Kyiv still doesn't feel like a city in crisis.
The government here is telling people to stay calm and united and, in the words of a statement this morning, refrain from actions that that undermine stability and sow panic. President Zelensky said the country had to be ready for any eventuality.
All across the country, foreign citizens are now making hurried plans. Stuart McKenzie, who's lived in Kyiv for 28 years and runs a successful business, hopes to get his wife and two sons out on a flight. But he is ready, if necessary, to pack the family into the car and drive 300 miles to Poland. He loves Ukraine and can't quite believe it's come to this.
At the British Embassy, we found tight-lipped staff loading bags into a car and driving off. No-one seemed keen to talk.
Not far to the north, across the border in Belarus, Russia's war games are now well under way. Russian defence ministry pictures, released this morning, showed multiple rocket launchers being fired. Moscow still says it has no plans to invade. But there's a lot the Russians can do without ever setting foot inside Ukraine.
In Kyiv, meanwhile, several thousand people marched through the city on Saturday, chanting slogans pledging loyalty to Ukraine and resistance to any Russian invasion. The march was organised by a right-wing nationalist group called Gonor and anti-Zelensky far-right activist Sergiy Sternenko, but it attracted a wider crowd.
BBC reporter Eleanor Montague says the demonstration was not huge, but was the first significant manifestation of public feeling since tensions escalated, finishing at the symbolic Maidan Square.
Sasha Nizelska, who works as a nanny in Kyiv, told the BBC that she would resist a Russian attack with all means in her power. The sentiment was repeated by people of all age groups attending the demonstration, our reporter said.
Tensions have steadily increased as Russia has continued to deploy troops along Ukraine's eastern border. Russian troops are also staging military exercises in Belarus to the north, while naval exercises in the Sea of Azov in the south-east have led to accusations that Russia is blocking Ukraine's access to the sea.