The city has seen months of intense fighting – despite its strategic value being questioned.
Russian and Ukrainian forces are fighting in the streets of Bakhmut – but Russia does not control the eastern city, its deputy mayor has said.
Oleksandr Marchenko also told the BBC the remaining 4,000 civilians are living in shelters without access to gas, electricity or water.
Mr Marchenko said “not a single building” had remained untouched and that the city is “almost destroyed”,
Bakhmut has seen months of fighting, as Russia tries to take charge.
“There is fighting near the city and there are also street fights,” Mr Marchenko said.
Taking the city would be a rare battlefield success in recent months for Russia.
But despite that, the city’s strategic value has been questioned. Some experts say any Russian victory could be pyrrhic – that is, not worth the cost.
Thousands of Russian troops have died trying to take Bakhmut, which had a pre-war population of around 75,000. Ukrainian commanders estimate that Russia has lost seven times as many soldiers as they have.
Now, after fierce shelling, Russian forces and troops from the Wagner private army appear to have surrounded much of Bakhmut.
On Saturday, UK military intelligence said Russian advances in the northern suburbs have left the Ukraine-held section of the city vulnerable to Russian attacks on three sides.
Mr Marchenko accused the Russians of having “no goal” to save the city and that it wanted to commit “genocide of the Ukrainian people”.
“Currently there is no communication in the city so the city is cut out, the bridges are destroyed and the tactics the Russians are using is the tactic of parched land,” Mr Marchenko told the Today programme.
Earlier this week, Ukraine’s President Volodymyr Zelensky said that the situation in the area was becoming “more and more difficult” – although the Ukrainian military said it had repelled numerous attacks since Friday.
The commander of Ukraine’s ground forces, Col Gen Oleksandr Syrskyi, visited Bakhmut on Friday for meetings with local commanders.
“I believe we shouldn’t give any inch of our land to the enemy,” Mr Marchenko said. “We should protect our land, we should protect our people and we should protect the businesses that are on this land.
“And the reason why we shouldn’t give it to them is because it will be very hard to take it back, to regain the control after Russians capture it.”
Russia claimed the Donbas town of Soledar, about 10km (6.2 miles) from Bakhmut, in January following a long battle with the Ukrainian forces.
Soledar, too, was reportedly reduced to a wasteland of flattened buildings and rubble by the time the Ukrainian army retreated.
On Friday, President Zelensky stressed that artillery and shells were needed to “stop Russia”.
US Secretary of State Antony Blinken said the country’s latest package included high-precision Himars artillery rockets and howitzers “which Ukraine is using so effectively”.