Both pro-Russian separatists and Ukrainian government officials have claimed military gains in the battle for the airport in the rebel stronghold of Donetsk.
Shelling could be heard in many districts of the eastern city, where fighting continued on Friday, its city council said.
An artillery attack on Thursday in northern Donetsk killed a Swiss Red Cross worker.
The 38-year-old administrator was killed when a shell landed near the organisation's office in the city, the International Committee of the Red Cross said.
The Donetsk city council said five civilians were injured in Thursday's attacks.
"Indiscriminate shelling of residential areas is unacceptable and violates international humanitarian law," said Dominik Stillhart, director of operations for the international Red Cross.
The United Nations expressed concern over the surge in fighting and mounting civilian casualties.
The "tragic incidents underscore the fragility of the current ceasefire," UN Secretary General Ban Ki-moon said.
The separatists said on Friday that they had established control over most parts of the airport in the city of 1 million people, which is the largest in the region.
The "defence ministry" of the self-declared Donetsk People's Republic said the Ukrainian army remained in the building's parts that are underground, Russia's Tass news agency reported.
The Ukrainian government was adamant that the airport remained under its control.
"Donetsk airport remains under the control of the anti-terrorist forces," military spokesman Vladislav Seleznev said.
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