The deputy mayor said that Russian and Ukrainian forces are fighting in the streets of Bakhmut, but Russia does not control the eastern city.
Oleksandr Marchenko told the BBC that the remaining 4,000 civilians were living in shelters without access to gas, electricity or water.
Marchenko said that "not a single building remained" intact and that the city was "almost destroyed".
Bakhmut has seen months of fighting as Russia tries to take over
"There is fighting near the city, as well as fighting in the streets," Marchenko said.
Capturing the city would be a rare battlefield success in recent months for Russia.
But despite this, the city's strategic value has been called into question. Some experts say that any Russian victory could be very expensive, which means that it is not worth the cost.
Thousands of Russian soldiers died while trying to seize Bakhmut, which had a population of about 75,000 before the war. Ukrainian commanders estimate that Russia lost seven times as many soldiers as it did.
Now, after a heavy bombardment, the Russian forces and Wagner's private army seem to have surrounded much of Bakhmut.
On Saturday, British military intelligence said that the Russian advance in the northern suburbs had made the Ukrainian-controlled part of the city vulnerable to Russian attacks from three sides.
Marchenko accused the Russians of having "no aim" to save the city and of wanting to commit "genocide of the Ukrainian people".
"Currently there are no communications in the city, so the city is isolated, the bridges are destroyed, and the tactic the Russians are using is the dry land tactic," Marchenko told the TODAY program.
Earlier this week, Ukrainian President Volodymyr Zelensky said the situation in the region was becoming "increasingly difficult", although the Ukrainian military said it had repelled several attacks since Friday.
The commander of the Ukrainian Ground Forces, Col. Oleksandr Sersky, visited Pakhmut on Friday to meet with local leaders.
"I think we should not give an inch of our land to the enemy," said Marchenko. “We must protect our land, we must protect our people, and we must protect the businesses that are on this earth.
The reason why they did not give it to them is that it would be very difficult to take it back, to regain control after the Russians took control of it.
The Donbass town of Solidar, about 10 kilometers (6.2 miles) from Bakhmut, was claimed by Russia in January after a protracted battle with Ukrainian forces.
Solidar has reportedly turned into a wasteland of collapsed buildings and rubble as the Ukrainian army withdrew.
On Friday, President Zelensky stressed that artillery and missiles are needed to "stop Russia."
US Secretary of State Antony Blinken said the country's latest package included HIMARS high-precision artillery missiles and howitzers "which Ukraine is using very effectively."