A young local resident passes by a crater caused by a Russian strike in Kramatorsk, Ukraine, on Saturday. AP

After ending weeks of resistance by the last Ukrainian fighters in the strategic southeastern city of Mariupol, Russia is waging a major offensive in Luhansk, one of two provinces in Donbas.

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Russian-backed separatists already controlled swathes of territory in Luhansk and the neighbouring Donetsk province before the Feb. 24 invasion, but Moscow wants to seize the last remaining Ukrainian-held territory in Donbas.

"The situation in Donbas is extremely difficult," Ukrainian President Volodymyr Zelensky said in his nightly address. The Russian army was trying to attack the cities of Sloviansk and Sievierodonetsk, but Ukrainian forces were holding off their advance, he said.

Smoke rises above a residential area near the town of Izium, Kharkiv, Ukraine, on Saturday. Reuters

Zelensky adviser Mykhailo Podolyak ruled out agreeing to a ceasefire and said Kyiv would not accept any deal with Moscow that involved ceding territory. Making concessions would backfire on Ukraine because Russia would hit back harder after any break in fighting, he said.

"The war will not stop (after concessions). It will just be put on pause for some time," Podolyak, Ukraine's lead negotiator, told Reuters in an interview in the heavily guarded presidential office. "They'll start a new offensive, even more bloody and large-scale."

Recent calls for an immediate ceasefire have come from US Defence Secretary Lloyd Austin and Italian Prime Minister Mario Draghi.

The end of fighting in Mariupol, the biggest city Russia has captured, gives Russian President Vladimir Putin a rare victory after a series of setbacks in nearly three months of combat.

Service members of the Ukrainian armed forces travel on a military vehicle near the town of Bakhmut. Reuters

The last Ukrainian forces holed up Mariupol's vast Azovstal steelworks surrendered on Friday, Russia said.

Full control of Mariupol gives Russia command of a land route linking the Crimean Peninsula, which Moscow seized in 2014, with mainland Russia and areas of eastern Ukraine held by pro-Russia separatists.

Ukrainian forces in the separatist-controlled regions of Luhansk and Donetsk said on Saturday they had repelled nine attacks and destroyed five tanks and 10 other armoured vehicles in the previous 24 hours.

Russian forces were using aircraft, artillery, tanks, rockets, mortars and missiles along the entire front line to attack civilian structures and residential areas, the Ukrainians said in a Facebook post. At least seven people had been killed in the Donetsk region, they said.

People walk past a destroyed building in Mariupol, Ukraine, on Saturday. AP

Russian troops destroyed a bridge on the Siverskiy Donets River between Sievierodonetsk and Lysychansk, Luhansk regional governor Serhiy Gaidai said. There was fighting on the outskirts of Sievierodonetsk from morning through the night, he said on the Telegram messaging app.

Sievierodonetsk and its twin Lysychansk across the Siverskiy Donets River form the eastern part of a Ukrainian-held pocket that Russia has been trying to overrun since mid-April after failing to capture Kyiv.

Reuters