Ukrainian President Volodymyr Zelenskyy said early on Sunday the siege of the port city of Mariupol would go down in history for what he said were war crimes committed by Russian troops.

“To do this to a peaceful city, what the occupiers did, is a terror that will be remembered for centuries to come,” Zelenskyy said in a video address to the nation.

Russian forces have pushed deeper into the besieged and battered city, where heavy fighting shut down a major steel plant and local authorities pleaded for more Western help.

In the capital, Kyiv, at least 20 babies carried by Ukrainian surrogate mothers are stuck in a makeshift bomb shelter, waiting for parents to travel into the war zone to pick them up.

Some just days old, the babies are being cared for by nurses who cannot leave the shelter because of constant shelling by Russian troops who are trying to encircle the city.

The fall of Mariupol, the scene of some of the war's worst suffering, would mark a major battlefield advance for the Russians, who are largely bogged down outside major cities more than three weeks into the biggest land invasion in Europe since World War II.

“Children, elderly people are dying. The city is destroyed and it is wiped off the face of the earth,” Mariupol police officer Michail Vershnin said from a rubble-strewn street in a video addressed to Western leaders that was authenticated by The Associated Press.

Details also began to emerge Saturday about a rocket attack that killed as many as 40 marines in the southern city of Mykolaiv the previous day, according to a Ukrainian military official who spoke to The New York Times.

Russian forces have already cut Mariupol off from the Sea of Azov, and its fall would link Crimea, which Russia annexed in 2014, to eastern territories controlled by Moscow-backed separatists.

It would mark a rare advance in the face of fierce Ukrainian resistance that has dashed Russia's hopes for a quick victory and galvanized the West.

Ukrainian and Russian forces battled over the Azovstal steel plant in Mariupol, Vadym Denysenko, adviser to Ukraine's interior minister, said.

“One of the largest metallurgical plants in Europe is actually being destroyed,” Denysenko said in televised remarks.

The Mariupol city council claimed hours later that Russian soldiers had forcibly relocated several thousand city residents, mostly women and children, to Russia.

It didn't say where, and AP could not immediately confirm the claim.

Zelenskyy adviser Oleksiy Arestovych said the nearest forces that could assist Mariupol were already struggling against “the overwhelming force of the enemy” and that “there is currently no military solution to Mariupol.”

Despite the siege in Mariupol, many remained struck by Ukraine's ability to hold back its much bigger, better-armed foe.

The United Kingdom's Defense Ministry said Ukraine's airspace continued to be effectively defended.

“Gaining control of the air was one of Russia's principal objectives for the opening days of the conflict and their continued failure to do so has significantly blunted their operational progress," the ministry said on Twitter.

Russia is now relying on stand-off weapons launched from the relative safety of Russian airspace to strike targets within Ukraine, the ministry said.

In Mykolaiv, rescuers searched the rubble of the marine barracks that was destroyed in an apparent missile attack Friday. The region's governor said the marines were asleep when the attack happened.

It wasn't clear how many marines were inside at the time, and rescuers were still searching the rubble for survivors the following day.

But a senior Ukrainian military official, who spoke to The New York Times on condition of anonymity to reveal sensitive information, estimated that as many as 40 marines were killed, which would make it one of the deadliest known attacks on Ukrainian forces during the war.

Estimates of Russian deaths vary widely, but even conservative figures are in the low thousands. Russia had 64 deaths in five days of fighting during its 2008 war with Georgia. It lost about 15,000 in Afghanistan over 10 years, and more than 11,000 in years of fighting in Chechnya.

Russia's number of dead and wounded in Ukraine is nearing the 10 per cent benchmark of diminished combat effectiveness, said Dmitry Gorenburg, a researcher on Russia's security at the Virginia-based CNA think tank.

The reported battlefield deaths of four Russian generals — out of an estimated 20 in the fight — signal impaired command, Gorenburg said.

Russia would need 800,000 troops — almost equal to its entire active-duty military — to control Ukraine long-term in the face of armed opposition, said Michael Clarke, former head of the British-based Royal United Services Institute, a defense think tank.

“Unless the Russians intend to be completely genocidal — they could flatten all the major cities, and Ukrainians will rise up against Russian occupation — there will be just constant guerrilla war,” said Clarke.

The Russian military said Saturday that it used its latest hypersonic missile for the first time in combat. Maj. Gen. Igor Konashenkov said Kinzhal missiles destroyed an underground warehouse storing Ukrainian missiles and aviation ammunition in the western region of Ivano-Frankivsk.

Russia has said the Kinzhal, carried by MiG-31 fighter jets, has a range of up to 2,000 kilometers (about 1,250 miles) and flies at 10 times the speed of sound.

Pentagon press secretary John Kirby said the US couldn't confirm the use of a hypersonic missile.

UN bodies have confirmed more than 847 civilian deaths since the war began, though they concede the actual toll is likely much higher. The UN says more than 3.3 million people have fled Ukraine as refugees.

Evacuations from Mariupol and other besieged cities proceeded along eight of 10 humanitarian corridors, Deputy Prime Minister Iryna Vereshchuk said, and a total of 6,623 people left.

Vereshchuk said planned humanitarian aid for the southern city of Kherson, which Russia seized early in the war, could not be delivered because the trucks were stopped along the way by Russian troops.

Ukraine and Russia have held several rounds of negotiations aimed at ending the conflict but remain divided over several issues, with Moscow pressing for its neighbour's demilitarisation and Kyiv demanding security guarantees.

Around Ukraine, hospitals, schools and buildings where people sought safety have been attacked.

At least 130 people survived the Wednesday bombing of a Mariupol theater that was being used a shelter, but another 1,300 were believed to be still inside, Ludmyla Denisova, the Ukrainian Parliament's human rights commissioner, said Friday.

“We pray that they will all be alive, but so far there is no information about them," Denisova told Ukrainian television.

A satellite image from Maxar Technologies released Saturday confirmed earlier reports that much of the theater was destroyed.

It also showed the word “CHILDREN” written in Russian in large white letters outside the building.

Russian forces have fired on eight cities and villages in the eastern Donetsk region in the past 24 hours, including Mariupol, Ukraine's national police said Saturday. Dozens of civilians were killed or wounded, and at least 37 residential buildings and facilities were damaged including a school, a museum and a shopping center.

In the western city of Lviv, Ukraine's cultural capital, which was hit by Russian missiles on Friday, military veterans were training dozens of civilians on how to handle firearms and grenades.

“It's hard, because I have really weak hands, but I can manage it,” said one trainee, 22-year-old Katarina Ishchenko.