Pensioner Tamara, 64, said she was going to stay with her sister in Zaporizhzhia. She was leaving with her husband, daughter, son-in-law and grandson.
“It is a pleasure … to leave after this nightmare. We lived in basements for 30 days,” she said tearfully.
A Ukrainian marine commander, Serhiy Volny, said fighters at the steel works may not be able to hold out for much longer. President Volodymyr Zelensky has said an estimated 1,000 civilians are sheltering there.
Ukraine was ready for a “special round of negotiations” with no conditions “to save our guys … military, civilians, children, the living and the wounded”, negotiator Mykhailo Podolyak said on Twitter.
Kyiv has proposed swapping Russian prisoners of war for safe passage for trapped civilians and soldiers. It was not known if Russia had responded to the offer of special negotiations.
Fighters remain holed up in the plant and have ignored an ultimatum by Russia to surrender.
Elsewhere in the east, Russia’s forces had carried out strikes on dozens of military facilities and had shot down a Ukrainian Mi-8 helicopter near the village of Koroviy Yar, its defence ministry said.
British military intelligence said Russian forces were keen to demonstrate significant success by May 9, known as Victory Day in Russia as it commemorates the surrender of Nazi Germany in 1945. Putin’s air force was providing close support for the offensive in the east where Russian forces were advancing towards the town of Kramatorsk.
Analysts say the offensive in the east could devolve into a grim war of attrition as Russia runs up against Ukraine’s most experienced, battle-hardened troops, who have been fighting pro-Moscow separatists in the Donbas for the past eight years.
Putin said the first test launch on Wednesday of Russia’s Sarmat intercontinental ballistic missile, a new and long-awaited addition to its nuclear arsenal, would “provide food for thought for those who, in the heat of frenzied aggressive rhetoric, try to threaten our country”.
The Pentagon described the test as “routine” and said it wasn’t considered a threat.
Russia calls its incursion a “special military operation” to demilitarise and “denazify” Ukraine. Kyiv and its Western allies reject that as a false pretext for a war of choice.
The Spanish and Danish Prime Ministers were due to visit Zelensky in Kyiv in a show of support on Thursday, following in the wake of UK Prime Minister Boris Johnson and the leaders of Poland, Slovenia and the Czech Republic.
US President Joe Biden is set to deliver an update on a new arms package for Ukraine, which is likely to be a similar size to an $US800 million one ($1000) announced last week, a US official said.
Mariupol, once a prosperous seaside city of 400,000, is now a wasteland where corpses lie in the streets as Russia slams the Azovstal steel plant with bunker-buster bombs, the government in Kyiv said.
The deputy commander of the Ukrainian Azov Regiment in Mariupol, Svyatoslav Kalamar, said several bunkers under the plant still held about 80-100 civilians each.
A Ukrainian posted a video plea on Facebook urging world leaders to help evacuate people from the plant, saying: “We have more than 500 wounded soldiers and hundreds of civilians with us, including women and children.”
The officer, who identified himself as Serhiy Volynskyy of the 36th Marine Brigade, said: “This may be our last appeal. We may have only a few days or hours left.” The authenticity of the video could not be independently verified.