The Donetsk ice hockey stadium was set on fire on Tuesday, reportedly after an armed group broke in
At least 30 pro-Russia separatists have been killed so far in fighting at the airport in the eastern Ukrainian city of Donetsk, insurgents say.
Armed separatists tried to take over the airport on Monday.
A BBC team near the airport on Tuesday said sporadic gunfire continued, though one Ukrainian military spokesman said it now controlled the facility.
New President Petro Poroshenko vowed on Monday "anti-terrorist operations" in the east would "last hours not months".
A representative of the self-proclaimed Donetsk People's Republic told the BBC that reports of at least 30 dead in Monday's fighting were accurate.
One insurgent told Associated Press the bodies had been brought to a hospital in the city.
Interior Minister Arsen Avakov said there were no casualties among Ukrainian troops.
Monday's clashes started after separatist militants stormed the Sergei Prokofiev Donetsk airport in the early hours of the morning.
The Ukrainian military responded quickly with air strikes and an assault by heavily armed troops.
Reporters said there was heavy gunfire throughout the day and night, with black smoke rising into the air.
The attempt to seize the airport may have been intended to prevent Poroshenko from travelling there after he said his first trip in office would be to visit the restive east.
Poroshenko, a 48-year-old billionaire and former foreign minister, was on Monday formally declared the winner of Sunday's presidential election with 54% of the vote.
He vowed east Ukraine would not be "turned into Somalia", adding: "The anti-terrorist operation cannot and should not last two or three months. It should and will last hours."
However, Poroshenko has also said he wants to talk to Russia to end the crisis.
Russia's Foreign Minister Sergei Lavrov said Moscow was "open to dialogue" with Poroshenko but insisted that military action against separatists must end.
Ukraine's military is engaged in an offensive in the east to tackle armed separatists who declared independence in the provinces of Luhansk and Donetsk.
The insurgency developed after pro-Moscow President Viktor Yanukovych fled Kiev amid bloody street protests and calls for closer ties with the EU.
Russia then annexed the Crimean peninsula in southern Ukraine.
At least two civilians were killed in Donetsk city and the separatist stronghold of Sloviansk on Monday.
In Sloviansk, the bloodied body of another woman lay near an apartment block. At least one man was also wounded.
Local people showed shell fragments and a fresh crater to an AFP news agency photographer.