A local official said 35 people were killed and 134 wounded in the March 13 attack on the sprawling Yavoriv International Centre for Peacekeeping and Security, a training base just 25 kilometers from the Polish border, bringing the conflict to the doorstep of the Western security alliance.

Regional governor Maksym Kozytskiy said Russian planes fired around 30 rockets at the facility, adding that some were intercepted before they hit.

While Western nations have sought to isolate Russian President Vladimir Putin by imposing harsh economic sanctions and have been supplying Ukraine with weapons, the United States and its allies hope to avoid NATO being drawn into the conflict.

Britain warned that the incident marked a "significant escalation" of the conflict while White House national-security adviser Jake Sullivan, speaking on CBS's "Face the Nation", said any attack on NATO territory would trigger a full response by the alliance.

President Joe Biden “has been clear, repeatedly, that the United States will work with our allies to defend every inch of NATO territory and that means every inch,” Sullivan said.

The attack highlighted the intensification of Russia’s assault on Ukraine, with heavy fighting reported in many areas across the country.

Amid the fighting, Ukrainian officials said Russia had agreed to open more than 10 humanitarian corridors on March 13, including from the besieged port city of Mariupol, where the city council said 2,187 people have been killed since the invasion started on February 24.

The International Committee of the Red Cross said on March 13 that hundreds of thousands of Mariupol’s residents are “facing extreme or total shortages of basic necessities like food, water and medicine.”

“Dead bodies, of civilians and combatants, remain trapped under the rubble or lying in the open where they fell,” it said in a statement.

“In the name of humanity, this cannot continue,” Peter Maurer, president of the ICRC added in a tweet.

The United Nations said on March 13 that nearly 2.7 million Ukrainians have fled to neighboring countries during the conflict.

"Today is the largest rally in Kherson! In the eyes of the occupiers there is despair, they hide behind balaclavas and look away. Yes, they have weapons, but we are morally stronger," a post on a Twitter account set up for videos from the city to be posted reads.

In Russia, thousands of people gathered and shouted anti-war slogans despite a heavy police presence and threats of arrests from security officials.

According to OVD-Info, an NGO that monitors arrests during protests, 866 people were arrested as anti-war protests erupted in at least three dozen cities in Russia.

Almost 15,000 people have now been arrested in Russia for protesting against President Vladimir Putin's decision to invade Ukraine,

Ukrainian President Volodymyr Zelenskiy, meanwhile, again warned Russian forces that they face a fight to the death if they try to occupy the capital, Kyiv, whose residents woke again to the sound of air raid sirens.

"If they decide to carpet bomb and simply erase the history of this region...and destroy all of us, then they will enter Kyiv. If that's their goal, let them come in, but they will have to live on this land by themselves," Zelenskiy said on March 13.

Despite the intense fighting, Russian and Ukrainian negotiators signaled some progress in talks to resolve the crisis on March 13.

"We will not concede in principle on any positions. Russia now understands this. Russia is already beginning to talk constructively," Ukrainian negotiator and presidential adviser Mykhaylo Podolyak said in a video posted online.

"I think that we will achieve some results literally in a matter of days," he said.

Leonid Slutsky, a Russian negotiator was quoted as saying by the RIA news agency that the talks had made substantial progress.

"According to my personal expectations, this progress may grow in the coming days into a joint position of both delegations, into documents for signing," Slutsky said.

Neither side said what the scope of any agreement might be.

In separate statements, Podolyak and Kremlin spokesman Dmitry Peskov said that the talks were scheduled to resume by videoconference on March 14.

Three rounds of talks between the two sides in Belarus had focused mainly on humanitarian issues and led to the limited opening of some corridors for civilians to escape fighting.