Ukraine says there is no question of surrendering the city of Mariupol after Russia called on Ukrainian forces to lay down arms in the besieged port city.

“There can be no question of any surrender, laying down of arms,” the Ukrainska Pravda news portal cited Ukraine’s Deputy Prime Minister Iryna Vereshchuk as saying on Sunday.

“We have already informed the Russian side about this.”

Russia said a “terrible humanitarian catastrophe” was unfolding in Mariupol.

“Lay down your arms,” Colonel-General Mikhail Mizintsev, the director of the Russian National Centre for Defence Management, said in a briefing distributed by the defence ministry.

“A terrible humanitarian catastrophe has developed,” General Mizintsev said.

“All who lay down their arms are guaranteed safe passage out of Mariupol.”

The Russian Ministry of Defence gave Mariupol authorities until 4am Monday (local time, 1pm AEDT) to surrender, Russian state media said.

Mariupol has suffered some of the heaviest bombardment since Russia invaded Ukraine on February 24. Many of its 400,000 residents remain trapped in the city with little if any food, water and power.

It came amid reports the capital, Kyiv, was also under heavy bombing early on Monday. Kyiv’s mayor and the local police have posted images of explosions in the Podilskyi district of the Ukraine capital to Telegram.

“Enemy shelling” caused a fire on several floors of a shopping centre, and set cars ablaze, emergency services said on Facebook.

“According to the information we have at the moment, several homes and one of the shopping centres [were hit],” mayor Vitali Klitschko said on his Telegram channel.

Local emergency services said at least four people had died.

Elsewhere, Ukraine Ministry of Foreign Affairs spokesperson Oleg Nikolenko called Russia’s actions in Mariupol on Sunday “a chapter from WWII”.

“First they came to destroy the cities, bombing hospitals, theatres, schools, and shelters, killing civilians and children. Then they forcibly relocated the scared, exhausted people to the invader’s land. A chapter from WWII? No – the actions of the Russian army, today in Mariupol,” Mr Nikolenko wrote.

The Mariupol city council said on its Telegram channel late on Saturday that thousands of residents had been “deported” to Russia in the past week.

General Mizintsev denied that, saying only that said Russia had evacuated 59,304 people out of the city. He said 130,000 civilians remained as effective hostages.

General Mizintsev also said 330,686 people had been evacuated from Ukraine by Russia since the start of its “operation”. He denied Russia was using heavy heavy weapons in Mariupol and said humanitarian corridors for civilians would open eastwards and westwards out of the city at 10am on Monday.

Pyotr Andryushenko, an adviser to the mayor of Mariupol, repeated the claims of forced evacuations, saying Moscow’s humanitarian promises could not be trusted.

“We will fight until the last of our soldiers,” Mr Andryushenko told the BBC.

“When they [Russian forces] say about humanitarian corridors, what do they really do? They really force evacuate our people to Russia.”

Ms Vereshchuk said Russia’s actions were “a deliberate manipulation”.

“Instead of spending time on eight pages on letters, just open the corridor,” she said.

Russia and Ukraine have traded blame for the failure to open such corridors in recent weeks.

General Mizintsev, without providing evidence, said that Ukrainian “bandits”, “neo-Nazis” and nationalists had engaged in “mass terror” and gone on a killing spree in the city.

Ukraine said it was fighting for its existence and President Volodymyr Zelensky said on Saturday the siege of Mariupol was “a terror that will be remembered for centuries to come”.

Russian President Vladimir Putin said the “special military operation” in Ukraine was necessary to disarm and “denazify” its neighbour. Russia has repeatedly denied it is targeting civilians.

The West has imposed sweeping sanctions on Russia that the Kremlin says amount to a declaration of economic war by the US and its allies.

Ukraine and its Western backers say Russian ground forces have made few advances in the last week, concentrating instead on artillery and missile strikes.

The United Nations refugee agency said 10 million people had been displaced across Ukraine, including some 3.4 million who have fled to neighbouring countries such as Poland. Officials in the region said they were reaching capacity to comfortably house refugees.

The UN human rights office said at least 902 Ukrainian civilians had been killed as of midnight Saturday. However, it says the real toll is probably much higher.

Ukrainian prosecutors said 112 children had been killed.

In the southern city of Kherson, a video clip obtained by Reuters showed dozens of protesters, some wrapped in Ukraine’s blue and yellow national flag, chanting “Go home” in Russian at two military vehicles bearing Russian markings. The vehicles turned and left the area.

Kyiv and Moscow reported some progress last week toward a political formula that would guarantee Ukraine’s security, while keeping it outside NATO – a key Russian demand. However, each side has accused the other of dragging things out.

Russian forces have also taken heavy losses, and long columns of troops that bore down on the capital Kyiv have been halted in the suburbs. Ukraine’s military said on Sunday that Moscow’s combat losses included 14,700 personnel and 476 tanks.

In an interview with CNN, Mr Zelensky reiterated that he was ready for talks with Mr Putin and that the war would not end without negotiations.

-with AAP