Volodymyr Zelenskiy has pleaded with the international community to do more as sanctions announced so far are not enough.
According to him, missiles on Friday pounded the Ukrainian capital as Russian forces pressed their advance.
Air raid sirens wailed over the city of three million people, where some were sheltering in underground metro stations, a day after Russian President Vladimir Putin launched an invasion that had shocked the world.
A Ukrainian official said a Russian plane had been shot down and crashed into a building.
A senior Ukrainian official said Russian forces would enter areas just outside the capital, Kyiv, later on Friday, adding that Ukrainian troops were defending positions on four fronts in spite of being outnumbered.
Meanwhile, an estimate of 100,000 people flew as explosions and gunfire rocked major cities.
Dozens have been reported killed, Russian troops seized the Chernobyl former nuclear power plant north of Kyiv as they advanced on the city from Belarus.
U.S. and Ukrainian officials said Russia aims to capture Kyiv and topple the government, which Putin regards as a puppet of the United States.
Zelenskiy said he understood Russian troops were coming for him but vowed to stay in Kyiv.
“The enemy has marked me down as the number one target,’’ Zelenskiy said in a video message. “My family is the number two target. They want to destroy Ukraine politically by destroying the head of state.’’
“I will stay in the capital. My family is also in Ukraine.’’
Russia launched its invasion by land, air and sea on Thursday following a declaration of war by Putin, in the biggest attack on a European state since World War Two.
Putin said Russia is carrying out a special military operation to stop the Ukrainian government from committing genocide, an accusation the West calls a baseless fabrication.
He said that Ukraine is an illegitimate state whose land historically belongs to Russia, a view which Ukrainians see as an attempt to erase their more than thousand year history.
Putin’s aim remain obscure, he has said he does not plan a military occupation, only to disarm Ukraine and remove its leaders.
However, having told Ukrainians that their state is illegitimate, it is hard to see how he could simply impose a new leader and withdraw.
Russia has floated no name of a figure it would regard as acceptable and none has come forward.
Meanwhile, Britain said Moscow’s aim was to conquer all of Ukraine, and its military had failed to meet its main objectives on the first day because it failed to anticipate Ukrainians would resist.
“Its definitely our view that the Russians intend to invade the whole of Ukraine,’’ Defence Secretary Ben Wallace told Sky.
“Contrary to great Russian claims and indeed President Putin’s sort of vision that somehow the Ukrainians would be liberated and would be flocking to his cause he’s got that completely wrong, and the Russian army has failed to deliver on day one, its main objective.’’
Ukrainians were circulating an unverified recording on Friday of a Russian warship ordering Ukrainian troops at a Black Sea outpost to surrender.
Ukrainian forces downed an enemy aircraft over Kyiv early on Friday, which then crashed into a residential building and set it ablaze, said Anton Herashchenko, an adviser to the interior minister.
It was unclear whether the aircraft was manned or whether it could be a missile.
Kyiv municipal authorities said at least eight people were injured when the object crashed into an apartment block.
“Horrific Russian rocket strikes on Kyiv,’’ Foreign Minister Dmytro Kuleba wrote on Twitter. “Last time our capital experienced anything like this was in 1941 when it was attacked by Nazi Germany.’’
Authorities said intense fighting was under way in the city of Sumy in the northeast.
A border post in the southeast had been hit by missiles, causing deaths and injuries among border guards, and air raid sirens sounded over the city of Lviv in the west of the country.
Asked if he was worried about Zelenskiy’s safety, U.S. Secretary of State Antony Blinken told CBS: “To the best of my knowledge, President Zelenskiy remains in Ukraine at his post, and of course we’re concerned for the safety of all of our friends in Ukraine.’’