Seven others have been injured.
The stampede at the event on Saturday organised by the Kings Assembly Pentecostal church in Rivers state involved people who came to the church's annual "Shop for Free" charity program, according to Grace Iringe-Koko, a police spokeswoman.
Such events are common in Nigeria, Africa's largest economy, where more than 80 million people live in poverty, according to government statistics.
Saturday's charity program was supposed to begin at 9 am but dozens arrived as early as 5 am to secure their place in line, Iringe-Koko said. Somehow the locked gate was broken open, creating a stampede, she said.
Godwin Tepikor from Nigeria's National Emergency Management Agency said first responders were able to evacuate the bodies of those trampled to death and bring them to the morgue. Security forces cordoned off the area.
Dozens of residents later thronged the scene, mourning the dead and offering any assistance they could to emergency workers.
Doctors and emergency workers treated some of the injured as they lay in the open field.
Videos from the scene showed the clothing, shoes and other items meant for the beneficiaries.
One witness who only identified himself as Daniel said "there were so many children" among the dead.
Five of the dead children were from one mother, he told the AP, adding that a pregnant woman also lost her life.
Some church members were attacked and injured by relatives of the victims after the stampede, according to witness Christopher Eze. The church declined to comment on the situation.
The police spokeswoman said the seven injured were "responding to treatment."
The "Shop for Free" event was suspended while authorities investigated how the stampede occurred.
Nigeria has recorded similar stampedes in the past.
Twenty-four people died at an overcrowded church gathering in the southeastern state of Anambra in 2013, while at least 16 people were killed in 2014 when a crowd got out of control during a screening for government jobs in the nation's capital, Abuja.
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