Volunteers help people arriving from Ukraine, at Berlin's central station in Berlin, Germany. File/Reuters

UNHCR, the UN's refugee agency, recorded 2,011,312 refugees on its dedicated website, 276,244 more than the previous count on Monday.

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UNHCR chief Filippo Grandi called it a "shocking milestone".

"Behind the monolithic statistics are two million stories of separation, anguish, and loss," he said.

People wait inside the main hall of the railway station in Przemysl, Poland. File/Reuters

Families have been "senselessly ripped apart", plunged into "despair and unimaginable suffering" by the "brutal war", he said.

Authorities and the UN expect the flow to intensify as the Russian army advances deeper into Ukraine, particularly as it approaches the capital, Kyiv.

Before Russia invaded, more than 37 million people lived in Ukrainian territory under the control of the central government.

Besides those who have left, an unknown number have been displaced from their homes within the country.

The International Organization for Migration said that 103,000 third-country nationals were among those who have fled.

"There are countless tens of thousands of others who remain in the country stranded," IOM spokesman Paul Dillon said, citing a mixture of overseas students and workers.

He announced a partnership with Airbnb to connect refugees to free or heavily discounted short-term housing in neighbouring countries, with more than 26,000 hosts having signed up so far.

Agence France-Presse