Russian air strikes killed 35 people at a military base outside Ukraine's western city of Lviv, local officials said on Sunday, in an attack that brings the conflict dangerously close to the Polish border.

Another nine people were also killed in a strike on the southern city of Mykolaiv, the regional governor said, while the capital Kyiv braced for possible encirclement by Russian forces.

In a video address posted on social media late Saturday night, President Volodymyr Zelensky was adamant that the Russians would not take Ukraine.

“The Russian invaders cannot conquer us. They do not have such strength.

They do not have such spirit. They are holding only on violence. Only on terror,” he said.

For the first two weeks following its February 24 invasion, Russia's forces had focused on eastern and southern areas of Ukraine, notably the strategic and heavily besieged port of Mariupol.

In recent days they moved to the centre, striking the city of Dnipro, and now the west, edging close to the frontier with EU and Nato member Poland.

Overnight, missiles struck a military training ground in Yavoriv near Lviv, about 20 kilometres (12 miles) from the Polish border, regional governor Maxim Kozitsky said.

He said 35 people were killed and 134 injured in the attack on the base, which was a training centre for Ukrainian forces with foreign instructors, including from the United States and Canada.

In the Black Sea city of Mykolaiv, near the strategic port of Odessa, regional governor Vitaliy Kim said nine people were killed in a Russian air strike.

The city of around 500,000 has been under attack by Russian troops for days and an AFP reporter said a cancer treatment hospital and an eye clinic there came under fire Saturday.

Meanwhile, efforts continue to get help to the strategic southern port city of Mariupol, which aid agencies say is facing a humanitarian catastrophe.

A convoy of aid headed for Mariupol was blocked at a Russian checkpoint, but hoped to arrive on Sunday, Ukraine's Deputy Prime Minister Iryna Vereshchuk said.

Ukraine says more than 1,500 civilians have died in a near two-week siege, which has left the city without water or heat, and running out of food.

Attempts to evacuate hundreds of thousands of people have repeatedly failed.

“Mariupol is still surrounded... Since they cannot bring down the Ukrainian army, they target the population,” a French military source said.

A top Russian officer described the situation in stark language.

“Unfortunately, the humanitarian situation in Ukraine is continuing to deteriorate rapidly, and in some cities, it has reached catastrophic proportions,” said the head of the Russian National Defence Control Centre, Mikhail Mizintsev.

'Relentless defence'

The Russians have advanced far enough to raise fears of Kyiv becoming encircled imminently.

Other cities have already fallen or been surrounded, with civilians targeted in what the United Nations warned could amount to war crimes.

In his video address, Zelensky appealed for more aid.

“I keep reiterating to our allies and friends abroad; they have to keep doing more for our country, for Ukrainians and Ukraine. Because it is not only for Ukraine, but it is for all of Europe,” he said.

He said “about 1,300” Ukrainian soldiers had been killed since February 24, giving his country's first official toll.

He claimed Russia had lost about 12,000 troops while Moscow, for its part, has only given a toll of 498 dead, released on March 2.

Also read: Five scenarios for Ukraine after invasion

At least 579 civilians have been killed, according to a tally by the United Nations, which stressed that its figures were probably much lower than reality.

The UN estimates that almost 2.6 million people have fled Ukraine since the invasion, most of them to Poland, in Europe's worst refugee crisis since World War II.

In Kyiv, only the roads to the south remain open and the city is preparing to mount a “relentless defence”, according to the Ukrainian presidency.

Kyiv mayor Vitali Klitschko said the capital was stockpiling food and medicine, while civilians are being brought in on buses from bombed-out suburbs.

Britain's ministry of defence estimated that Russian forces were about 25 kilometres from Kyiv on Saturday and that a column north of the city had dispersed, reinforcing the indication of an attempt to encircle it.

However, the Russians are encountering resistance from the Ukrainian army to both the east and west of the capital, according to AFP journalists on the spot.

At the demolished bridge between the northwest suburb of Irpin and Kyiv itself, an AFP reporter on Sunday saw Ukrainian troops carry the corpses of three fellow soldiers on stretchers across the river on a makeshift plank bridge.

Ukrainian soldiers said they believe the Russians have overestimated their resources, in terms of troops and equipment, and underestimated those of their opponents.

“They have to camp in villages in temperatures of nearly minus 10 Celsius at night. They lack provisions and have to raid houses,” said one soldier, Ilya Berezenko, 27.

Glimmer of hope

Efforts for a diplomatic solution to the conflict continued, with French President Emmanuel Macron and Germany's Olaf Scholz speaking to Russian counterpart Vladimir Putin again Saturday.

They urged him to end the deadly blockade of Mariupol, the French presidency said.

Facing growing international condemnation, Putin sought to turn the tables, slamming Kyiv for what he described as the “flagrant violation” of international humanitarian law and accusing Ukraine's army of executing dissenters and using civilians as hostages.

The French presidency denounced his accusations, made during the talks with Macron and Scholz, as “lies”.

But in a small glimmer of hope, Zelensky said Saturday that Russia had adopted a “fundamentally different approach” in the latest talks to end the conflict.

He told reporters it was in contrast to earlier talks at which Moscow only “issued ultimatums” and that he was “happy to have a signal from Russia”.

Putin said last week he saw “some positive shifts” in their near-daily dialogue.

As Russia widens its bombardment, Zelensky's pleas for help have grown increasingly desperate.

Washington and its EU allies have sent funds and military aid to Ukraine and taken action against Russia's economy and oligarchs. A cultural and sporting boycott has further isolated Moscow.

In Irpin on Saturday, a Ukrainian soldier who gave his name only as Viktor showed off his British anti-tank missile system and the twisted remains of a Russian vehicle it destroyed.

“I want to say a big thank you to our British comrades helping us,” he said.

As international sanctions against Moscow have steadily tightened, crippling Russia's economy, the country's space agency Roscosmos warned on Saturday that the International Space Station could crash if Russian spacecraft serving it are affected.

But Washington on Friday added more layers of sanctions, this time ending normal trade relations and announcing a ban on Russian vodka, seafood and diamonds.

And on Saturday, US President Joe Biden authorised up to $200 million in new weapons and other aid to Ukraine.

But he has ruled out direct action against nuclear-armed Russia, warning that it would lead to “World War III”.