Russian forces on Thursday laid siege to urban areas across Ukraine in a grinding offensive that has pummelled civilian neighbourhoods with increasingly heavy artillery and reduced basic services to rubble.

The Russian assaults a day after gaining control of their first major Ukrainian city have deepened a humanitarian crisis that has prompted one million people to flee the country, and a million more to abandon their homes for Ukrainian cities farther from the fighting.

The UN said that a million Ukrainians had become refugees in the week since Russia launched its invasion — one of the largest exoduses in recent times. Many more have left their homes and are seeking safety in the western reaches of Ukraine or taking shelter underground.

A Ukrainian negotiator said on Thursday that after a second round of ceasefire talks with Russia, the two sides had reached an understanding on creating humanitarian corridors to evacuate civilians.

Ukrainian presidential adviser Mykhailo Podolyak said the two sides envisaged a possible temporary ceasefire to allow for the evacuation of civilians. They had also reached an understanding on the delivery of medicines and food to the places where the fiercest fighting was taking place.

It was the first time the two sides had agreed any form of progress on any issue since Russia invaded Ukraine a week ago.

Podolyak said the outcome had fallen short of Kyiv’s hopes, however.

The swift fall of the major city of Kherson in the south of the country raised fears that other cities could soon follow as Russia’s bombardment created pressure on people to surrender.