Kyiv, Ukraine – Explosions and gunfire rocked Kyiv overnight, but by Saturday morning, Ukraine's government said the military had repelled Russian attacks and the situation in the capital was tense but "under control."

President Volodymyr Zelensky had warned Ukrainians that Russian troops would likely "storm" the city during the night and called on people to "stop the enemy wherever possible."

But fears of intense urban warfare did not come to pass. As the sun rose over Kyiv, Zelensky released a video in which he refuted reports suggesting he had fled the country and declared: "I'm still here."

After the night of ground and air attacks in the Kyiv area and elsewhere, he said Ukrainians would not lay down their weapons or surrender to the invading forces, as Russian President Vladimir Putin demanded.

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Mykhailo Podolyak, an advisor to Zelensky, said the Russians were still trying to move a large number of troops and equipment into the city and that fighting was taking place in outlying areas.

But Ukrainian troops were mounting a strong defense and "both in the city itself and on the outskirts of Kyiv, the situation is under control," Podolyak said in comments carried by Ukraine's UNIAN news agency.

Kyiv remained on edge as the Russians pressed ahead and authorities told people to hunker down in bomb shelters if they can and take precautions, like staying away from windows.

"Last night was difficult, but there are no Russian troops in the city," Mayor Vitali Klitschko said in a video published on the Telegram news channel on Saturday morning.

But he added: "The enemy is trying to advance into the city."

As of 5:00 AM local time, Kyiv had recorded 35 injured, including two children, Klitschko said.

Kyiv's infrastructure was still intact, and underground trains were reportedly running.

Ukraine's foreign minister posted an image of smoke coming out of a Kyiv high-rise that had at least four floors destroyed during fighting. It was not yet clear if there were casualties.

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"Kyiv, our splendid, peaceful city, survived another night under attacks by Russian ground forces, missiles," Foreign Minister Dmytro Kuleba wrote.

"I demand the world: fully isolate Russia, expel ambassadors, oil embargo, ruin its economy. Stop Russian war criminals!" he continued.

The Ukrainian army said it repelled an attack on a barracks in Kyiv's west, only about 4.5 miles from the city center. Photos showed the site on fire while the sound of gunfire was audible on videos shared on social media.

Earlier in the night, there were reports that Russian troops tried to attack a combined heat and power plant in the far north-east of the city. Fighting was also reported in the Obolon district in the north.

Both Russia and Ukraine released new figures on the war's toll in terms of lives and infrastructure, although the information could not be independently verified.

Russia has crippled the operations of more than 800 Ukrainian military infrastructure sites, including airfields, command posts, anti-aircraft missile systems, and radar stations, the Defense Ministry in Moscow said on Saturday, the third day of Russia's offensive.

Ukraine's Defense Ministry said some 3,500 Russian soldiers had been killed and 200 others captured.

In addition, it said 14 planes, eight helicopters and 102 tanks as well as more than 530 other military vehicles were destroyed.

Meanwhile, Ukrainians continued to try and escape the violence by crossing into neighboring countries.

The Polish deputy interior minister said on Friday night 35,000 people have crossed into the country in the course of just 24 hours.

A dpa reporter at the Medyka-Shegyni crossing reported that refugees, including many traveling by foot, were waiting for hours to be processed.

The United Nations said Friday that tens of thousands of people were fleeing to Ukraine's neighbors and that, if the war escalated, the number could potentially reach 4 million or more.

Outside of Kyiv, the Ukrainian Defense Ministry said Russian air attacks struck the north-eastern city of Sumy, Mariupol in the south, and Poltava in the east.

Meanwhile, Russian troops have landed near the western Ukrainian city of Lviv, according to Mayor Andriy Sadovyi.

Three helicopters had dropped off about 60 paratroopers some 56 miles north-east of Lviv, near the town of Brody, on Saturday morning, Sadovyi wrote on Facebook. "Our troops have repelled them," he added.

The attackers had retreated for the time being. "We are keeping the situation under control," Sadovyj said.

Russia said it captured the south-eastern Ukrainian town of Melitopol, near the Sea of Azoz, and that Moscow-backed separatists had made territorial gains in the Donbass region.