Orikhiv, southeast Ukraine — Ukraine claims to be advancing in the fierce, months-long battle for the eastern city of Bakhmut. The leader of Russia's Wagner mercenary group, which has led Moscow's effort to try to capture the industrial town, admitted that Ukrainian troops have made gains.

With his ground war struggling, Russian President Vladimir Putin's forces have intensified their aerial assault on Ukrainian cities ahead of a long-anticipated Ukrainian counteroffensive. President Volodymyr Zelenskyy has urged his people to have patience, saying Ukraine stands to lose a lot more lives if the offensive is launched too soon.

• U.K. giving Ukraine long-range cruise missiles for counteroffensive

In the meantime, Ukrainian civilians in towns all along the front line in the country's east continue to bear the brunt of Putin's assault. Only about three miles from Russian positions, Orikhiv bears all the scars of a battleground. The town sits squarely on the front line of this war, and the few residents who haven't already fled live in constant fear of Russian attack.

Above ground, Orikhiv has been reduced to a ghost town of shattered glass and destroyed buildings. But below street level, CBS News met Deputy Mayor Svitlana Mandrych, working hard to keep herself and her community together.

"Every day we get strikes," she said. "Grad missiles, rockets, even phosphorus bombs."

Mandrych said the bombardment has been getting much worse.  

"We can't hear the launch, only the strike," she said. "It's very scary for people who don't have enough time to seek cover."

The deputy mayor led our CBS News team to a school that's been turned into both a bomb shelter and a community center.

From a pre-war population of around 14,000, only about 1,400 hardy souls remain. The last children left Orikhiv three weeks ago, when it became too dangerous. Locals say the town comes under attack day and night, including rockets that have targeted the school.

Ukraine's government calls shelters like the one in Orikhiv "points of invincibility" — an intentionally defiant title. Like others across the country's east, it's manned by volunteers — residents who've decided to stay and serve other holdouts, despite the risks.

Mandrych said every time explosions thunder above, fear grips her and the others taking shelter. She said she was always scared "to hear that our people have died."

As she spoke to us, as if on cue, there was a blast.

"That was ours," she explained calmly. "Outgoing."

Hundreds of "points of invincibility" like the school offer front-line residents a place to not only escape the daily barrage, but also to weather power outages, to get warm and fed, even to grab a hot shower and get some laundry done. There's even a barber who comes once a week to offer haircuts.

Mandrych said it's more than just a little village within the town, however. The school is "like civilization within all of the devastation."

Valentyna Petrivna, among those taking shelter, said her house "no longer exists" after being bombed. But she told CBS News she wouldn't leave her hometown.

"I am not so worried — I am worried more about my children. My son is fighting, and my grandchildren are in Zaporizhzhia," she said, referring to the larger city nearby that's also under constant attack by Russia's forces.

The people defiantly holding out in Orikhiv share more than a hot drink and each other's company. They're united in defiance - and hope that the war will end soon, so families can be reunited.

The residents told CBS News that despite their town's perilous location on the front line, they can't wait for the counteroffensive to begin. They're desperate for Ukraine's troops to push the Russians back far enough that they lose interest in randomly bombing the neighborhoods of Orikhiv.

In:
• Wagner Group
• War
• Bakhmut
• Ukraine
• Russia
• Missile Launch
• Vladimir Putin
• Volodymyr Zelenskyy