KYIV, Ukraine — The Ukrainian defense forces, outmanned and outgunned, waged a ferocious resistance to the Russian invasion on Saturday, battling to keep control of the capital, Kyiv, and other cities around the country.

President Volodymyr Zelensky of Ukraine posted a video on Twitter, telling the public not to believe false reports.

He was alive. Kyiv had not fallen. Any reports of Ukraine laying down its arms was a lie, Mr. Zelensky said.

“I’m here,” he said. “We are not putting down any arms. We will protect our country, because our weapons are our truth. The truth is that this is our land, our country, our children, and we will protect them all.”

“That is it. That’s what I wanted to tell you. Glory to Ukraine.”

His comments, released before 9 a.m., came as fighting intensified in Kyiv. What until three days ago had been a thriving European metropolis has been transformed into a battle zone. Russian troops pressed in from all directions.

There was intense street fighting, and bursts of gunfire and explosions could be heard across the city, including its heart, Maidan square, where in 2014 Ukrainian protests led to the toppling of a pro-Moscow government.

The Russian military has a decisive edge in cyberwarfare, tanks, heavy weaponry, missiles, fighter planes and warships. In sheer numbers, its military dwarfs that of Ukraine.

Russia has established attack lines into three cities — Kyiv in the north, Kharkiv in the northeast and Kherson in the south — and Ukrainian troops are fighting to hold all three. The Pentagon reported late Friday that the Russians did not appear to be in control of a single major population center. Significantly, a senior U.S. defense official said, Ukrainian command and control remains intact.

The Ukrainian government reported that hundreds of Russian soldiers had been killed in the war, along with scores of their own soldiers, while the Russian defense ministry issued a statement on Saturday morning that made no mention of any casualties or anything about the fight for Kyiv.

The Russian invasion started with targeted airstrikes before dawn on Thursday, but on the third day of the war, bloody battles were often being waged in close quarters. Ukrainska Pravda, a Ukrainian news site, citing witnesses, reported combat 400 yards from Maidan Square in central Kyiv.

All Ukrainian men of fighting are being drafted into service, and tens of thousands are signing up. Ukrainians were asked to make Molotov cocktails. And there were tearful scenes at airports in western Ukraine as wives kissed their husbands goodbye before they headed to front.

The nation has rallied around its president, Mr. Zelensky, a former comedian.

To him and other officials, the objective of the Russian invasion of a neighboring country that possessed no military threat is to topple the government.

Mr. Zelensky has said that he is “target no. 1.”

As battles were waged around the city on Saturday morning, there were reports of clashes near the city’s train station and along a central thoroughfare, Bohdan Khmelnitsky Street, leading from Victory Square toward the city center, according to the witness accounts. Along that street, closer to the city center, bursts of gunfire could be heard through the night.

“We are stopping the horde, so far as we can,” the secretary of the Ukrainian Security and Defense Council, Oleksy Danilov, said around 7 a.m. “The situation is under control of the Armed Forces of Ukraine and citizens of Kyiv.”

In dozens of interviews in the tense hours before the invasion and in the days after, Ukrainians struggled to understand how a country at peace so suddenly found itself at war. For many Ukrainians, the answer was found in Russia’s president, Vladimir V. Putin.

This is Mr. Putin’s war. But what frightened people perhaps as much as the threat of missiles and bombs was that they did not know what he wanted.

The fear was evident in the drive from Kyiv to a small village outside the city. Military convoys had replaced families going on vacation or visiting friends. Where once Kyiv was known as a city where the music played a touch too loud in its cafes, the incessant wail of air raid sirens drowned out all joy.

The fear was evident in the faces of the people seeking safety in Western Ukraine, after they emerged from 20-hour train rides in packed carriages that were kept pitch-black to avoid being targeting by Russian rockets.

From Lviv in the west to Odessa in the south and Kharkiv and nearly all points in between, people huddled in air raid shelters and lined up and bank machines and stocked up on essentials.

While the Russians, for the moment, were not in control of any city, it was only the first phase of a conflict that could stretch into weeks or longer.

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