Russians were shelling many districts of Mykolaiv, the government said Wednesday. For the second day in a row, the town of Bereznehuvate came under fire Tuesday, according to regional authorities. Shelling in adjacent rural areas set fire to crops, they added.
The government said heavy fighting was raging in several villages along the regional border.
In Kherson, which has been under Russian control since March, more activists, politicians and journalists are reported to have been abducted.
"There is no Ukrainian media in the region," Ukrainian authorities said.
"The occupiers and local collaborators are making more and more loud statements about Kherson region joining Russia," the government said, but added that "every day more and more Ukrainian flags and inscriptions appear in the city (of Kherson)."
Some background: The extent of dissent and resistance in Kherson is difficult to gauge, but several attacks have been made on Ukrainian officials who chose to collaborate with the Russians, as well as poster campaigns against the occupation.
Earlier this week, Iryna Vereshchuk, Ukraine's Deputy Prime Minister, urged civilians in Kherson region to leave if they could.
She said that if necessary they should travel through Russian-annexed Crimea and added that it "is almost the only" evacuation corridor available to those wanting to flee.
Russian occupying forces have made it increasingly difficult for civilians to leave Kherson for Ukrainian-held territory.
Anecdotal evidence suggests hundreds of Ukrainians have left Kherson through Crimea, taking buses through Turkey or Russia and Georgia in a long trek to reach parts of Ukraine not under Russian control.
"According to our calculations, up to 50% of the region's population, which is half a million people have already left Kherson and Kherson region," Hennadii Lahuta, head of Kherson regional military administration, said Tuesday.
Routes out to Kryvyi Rih and Mykolaiv, which Kherson region residents previously used for evacuation, "do not work now, the occupiers do not let people out. There are columns which are being let out, but people are forced to spend weeks in the fields and go through occupied Melitopol and Vasylivka to Zaporizhzhia," he added.
More than 1,400 people had left occupied territories in the previous 24 hours, of whom about 400 had come from Kherson region and reached Zaporizhzhia, Oleksandr Starukh, head of that region's military administration, said Tuesday.