MAZYR, Belarus—A string of seven bus-size Russian military ambulances—their windows blocked with gray shades—pulled up to the back entrance of the main hospital here, about 30 miles from the border with Ukraine, on Tuesday evening, ferrying casualties from the front.

The convoy was part of what local residents and doctors said has in recent days become a steady flow of Russian soldiers wounded in fierce fighting around Kyiv, the Ukrainian capital, where a Russian advance has stalled in the face of strong resistance.

A doctor at the hospital—which is in southern Belarus’s Gomel region, a main staging ground for Russia’s offensive—said injured Russian troops began arriving on Monday. “I hope they don’t jail me for sharing this,” she said.

“I saw seven soldiers right in front of me out of the elevator. All Russian. One was in a wheelchair, the other had his head bandaged,” said a man leaving the hospital. “My doctor said they’ve closed off the whole surgical wing.”