The Jeong Woo 2 is still burning and appears to be sinking, the Rescue Coordination Centre New Zealand says.
Another 37 fishermen were rescued, but at least two of them are said to have serious burns.
The ship got into trouble in the Ross Sea, about 600km (375 miles) north of the US McMurdo Antarctic base.
Two other South Korean fishing vessels rushed to evacuate the crew members, after the 51-metre (167ft) Jeong Woo 2 issued a distress call early on Wednesday.
It adds that a US research vessel – which has onboard medical facilities – is “steaming north” to help the injured, as the two rescue boats remain stuck due to ice and fog conditions.
The incident happened about 3,700km (2,000 [nautical] miles) south-east of New Zealand.
The Jeong Woo 2 was built in Japan and is registered in Busan, South Korea, according to the Associated Press News Agency.
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