To the peal of church bells, and with more than a thousand mourners marching behind it, a coffin carrying the body of Mikhailo Zhyznevsky was borne through the streets of Kiev on Sunday. Zhyznevsky, a Belarusian who lived in Ukraine, was one of at least three victims of clashes between police and protesters last Wednesday. He died of gunshot wounds.

The sombre procession came as the three political leaders who have become the de facto leaders of the protests pondered a surprise offer of concessions extended to them on Saturday by the president, Viktor Yanukovych. As the tense standoff between protest barricades and riot police continued in Kiev, and unrest spread to the rest of the country, the president offered a wide range of concessions at negotiations with the leaders.

He promised Arseniy Yatsenyuk, of the jailed former prime minister Yulia Tymoshenko's Fatherland party, the prime minister's job, while the former boxer Vitali Klitschko was offered a deputy prime minister post. Yanukovych also suggested making a number of legislative and constitutional amendments during a special emergency parliament session on Tuesday.

The opposition leaders have reacted cautiously to the proposals, rejecting them but saying they are open for further negotiation. The mood on Independence Square is that nothing short of snap presidential elections will be enough to disperse the crowds from central Kiev and calm the tension, and the political leaders fear that if they do accept any other deal, they will lose the protest mandate, and harm their own political chances.

Yevgeniya, a 68-year-old from Kiev who was on Independence Square, said: "We know what this president is like. Power sharing won't work. It will be a trap. He has to go. Nothing else will do."

She said the violent clashes that broke out last week worried her, but were a sign of desperation: "They are young lads and of course they have hot heads. But it's the president's fault that it has come to this. It took people dying for him to listen to us."

In recent days, the problems for Yanukovych have intensified as the protest movement has spread from Kiev to more than a dozen regional cities, with crowds storming regional administration buildings and in some cases building barricades around them. On Sunday this new phenomenon spread to a number of cities in the east of the country, traditionally the president's stronghold.

There were also further clashes in Kiev itself during the early hours of Sunday morning, as protesters moved to storm the Ukraine House, a huge convention centre that was built to house a Soviet-era museum to Vladimir Lenin. A group of interior ministry troops were resting inside the building, and amid rumours that they were preparing to launch an attack on the protest barricades, the decision was taken to storm the building.

There were tense scenes as rows of well-organised youths, clad in makeshift body armour and wielding clubs, baseball bats and in one case a weaponised garden trowel advanced on the building. Fireworks were launched at the entrance, windows smashed, and some protesters attempted to lob molotov cocktails, but were restrained by fellow protesters. In the end, a human corridor was made for the troops to leave the building, which they did after several hours of tense standoff in which there were repeated rumours that the government might launch an attack to free the besieged conscripts. Klitschko arrived on the scene and negotiated the exit of the troops, who left the building at around 4am local time.

The huge conference hall was quickly occupied by the protesters and became another building in central Kiev under their control. On Sunday, the upper floors were occupied by sleeping protesters, while volunteers drove vans of food to the door and field kitchens were set up inside serving sandwiches and hot tea.

Down the street by the Dynamo Kiev football stadium, where the fatal clashes occurred last week, protesters have constructed huge barricades made of sandbags filled with snow, which grow bigger by the day. On Sunday there was a family atmosphere at the barricade, with groups of people, including grandmothers, banging on tin drums to create an infernal noise, and periodic shouts of: "Glory to Ukraine," and "Out with the gang!"

At the very frontline a few hardcore protesters stood guard, patrolling the soot-covered post-apocalyptic landscape, strewn with the smouldering carcasses of police buses, mangled metal and barbed wire. One of them, 34-year-old Sergei, had arrived two days earlier from the south of the country.

"I've read that people claim we are fascists," he said, referring to the hardline far-right groups who have indeed played a role in co-ordinating the violent part of the protest. "We aren't fascists. We are here to stand up for the future of our country. The fascists are over there," he said, waving a hand towards the lines of riot police standing around 20 metres away.

"My wife wouldn't let me come to Kiev before. She said it was too dangerous. But now I've decided it's more dangerous to stay sitting at home complaining, and not fight for a better future for my wife and kids. So now I'm here. If we retreat now, we will lose everything. There can't be any compromise. It's all over for Yanukovych."