The fire broke out around lunchtime on Friday and was contained late in the day, the safety ministry said.
Another 35 people suffered minor injuries, the fire authorities added.
They said the factory was owned by Anjun Industrial, which makes engine valves and is a supplier for Hyundai Motor and Kia Corp, among others, according to its website.
Reuters was unable to contact Anjun for comment.
In a statement on the company's website, CEO Sohn Ju-hwan apologised, adding it would fully co-operate with authorities, investigate the cause of the accident, review its safety systems and inspections, and swiftly implement all necessary measures to prevent a recurrence.
The government activated a centralised disaster management system after President Lee Jae Myung ordered all available resources to be used to deal with the fire and save lives.
The fire authorities said that one victim was found on the first floor, four on the second, and nine on the third.
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