A Russian ultimatum to Ukrainian troops in Mariupol to surrender or die has expired with no mass capitulation, but the commander of a unit believed to be holding out in the besieged city says his forces could survive just days or hours.

Thousands of Russian troops backed by artillery and rocket barrages were attempting to advance elsewhere in what Ukrainian officials call the Battle of the Donbas – a push by Moscow to seize two eastern provinces it claims on behalf of separatists.

In a video, the commander of Ukraine’s 36th Marine Brigade, one of the last units believed to be holding out in Mariupol, asked for international help to escape the siege.

“This is our appeal to the world. It may be our last. We may have only a few days or hours left,” Major Serhiy Volyna said in a video uploaded to Facebook.

“The enemy units are dozens of times larger than ours, they have dominance in the air, in artillery, in ground troops, in equipment and in tanks.”

Major Volyna, who has said women and children are trapped in cellars under the plant, spoke in front of a white brick wall in what sounded like a crowded room.

Reuters could not verify where or when the video was filmed.

The United Nations said on Wednesday the number of refugees who have fled Ukraine since Russia invaded on February 24 had surpassed five million. More than half of those are children.

Russia’s nearly eight-week invasion has so far failed to capture any of Ukraine’s largest cities.

Moscow was forced to retreat from northern Ukraine after an attack on Kyiv was repelled last month, but has poured troops back in for an assault on the east that began this week.

In the ruins of Mariupol, site of the war’s heaviest fighting and worst humanitarian catastrophe, Russia was hitting the last main Ukrainian stronghold, the Azovstal steel plant, with bunker-buster bombs, Kyiv said.

“The world watches the murder of children online and remains silent,” presidential adviser Mykhailo Podolyak wrote on Twitter.

Russia has been trying to take full control of Mariupol since the war’s first days.

Its capture would be a big strategic prize, linking territory held by pro-Russian separatists in the east with the Crimea region Moscow annexed in 2014.

Russian-backed separatists said shortly before a 2pm (9pm AEST) Wednesday deadline just five people had surrendered.

The previous day, Russia said no one had responded to a similar ultimatum.

Ukraine announced plans to send 90 buses to evacuate 6000 civilians from Mariupol, saying it had reached a “preliminary agreement” with Russia on a safe corridor, for the first time in weeks.

None of those earlier agreements have succeeded on the ground, with troops blocking all convoys.

The battle for the Donbas region, which includes the provinces of Luhansk and Donetsk, could be decisive as Russia searches for a victory to justify President Vladimir Putin’s invasion.

Russian television on Wednesday showed Mr Putin addressing a girl from Luhansk.

“It was the tragedy that took place in the Donbas, including in the Luhansk People’s Republic, that forced, simply forced Russia to launch this military operation, which everyone is well aware of today,” he said.

Peace talks have stalled. The Kremlin accused Kyiv of delaying the talks and changing its position, Kyiv said Moscow blocked progress by refusing humanitarian ceasefires.

Moscow is hoping its firepower advantage will bring more success than in the failed Kyiv campaign, when its overstretched supply lines were attacked by nimble small units.

Russian forces captured Kreminna, a frontline town of 18,000 people, on Tuesday.

Ukraine’s general staff said Russian forces had attempted an offensive near Kharkiv, the country’s second-biggest city, which is close to Russia’s supply lines to Donbas.

Charles Michel, head of the European Council that groups the 27 EU member states, arrived in Kyiv as the latest official to visit and demonstrate support.

-Reuters