Clashes in Nasir County, Upper Nile State, between forces loyal to President Salva Kiir and First Vice President Riek Machar have threatened to undermine their fragile peace-sharing agreement.
The attacks have drawn international concern, with the United Nations warning the escalating violence has displaced at least 50,000 people since February.
County commissioner James Gatluak said the assault began at around 3.30 am local time (0130 GMT), when the South Sudan army dropped multiple bombs on Nasir town's market and then a residential area.
One woman was wounded, he said, with a young child in critical condition following the assault.
"The people that are being affected most are the women and the children," he told AFP.
A Nasir resident, Jany Manytap, confirmed the attack and the wounded, adding: "The airstrike targets everyone."
Gatluak said the situation was "deteriorating" in Nasir, considered a stronghold of Machar's supporters, with humanitarian staff evacuated as the violence has escalated.
It follows another aerial attack on Nasir earlier this week that killed 20 people, according to Gatlauk.
Information Minister Michael Makuei Lueth said the strikes were part of "security operations", adding: "If you as a civilian happen to be there... then there is nothing we can do".
The fighting threatens a 2018 peace deal between Kiir and Machar, who fought a five-year civil war that killed some 400,000 people.
Kiir's allies have accused Machar's forces of fomenting unrest in Nasir County in league with the White Army, a loose band of armed youths from the vice president's Nuer ethnic community.
Tensions spiked earlier this month when an estimated 6,000 White Army combatants overran a military encampment in Nasir.
A rescue attempt by the United Nations led to the death of a UN crew member and senior South Sudanese general, among others.
Information Minister Lueth has also confirmed the presence of Ugandan forces in Juba on a "military pact", a week after denying their deployment to South Sudan.
The rising unrest has sparked international concern, with the head of the UN Mission in South Sudan, Nicholas Haysom, warning the country is "poised on the brink of relapse into civil war".
Analysts fear the unrest is being fuelled by the devastating nearly two-year civil war in neighbouring Sudan, with armed men flowing back and forth across the border.
"That is part of the biggest fear we are seeing now, where the two conflict systems could potentially merge," International Crisis Group's senior South Sudan analyst Daniel Akech told AFP.
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