One man drowned when the boat sank, while a woman and two children aged five and eight died after they were pulled from the water alive with 14 other people, according to the local authority of the Nord department.
Some of those rescued, including the two children, had gone into cardiac arrest afer being plucked from the water, while others, including more children, suffered hypothermia.
The group's small fishing boat capsized in the Channel at about 9:30 am, official Herve Tourmente told journalists earlier in Loon-Plage, near Dunkirk.
"It seems that one person, who might be an infant, is still missing," Tourmente said, noting that stormy conditions had made the attempted crossing especially perilous.
A passing English vessel reported that the migrant boat was in trouble, triggering a large search and rescue operation involving six boats with aerial support.
There were believed to have been about 20 migrants on board, meaning some may still be missing.
A growing number of migrants have tried to reach Britain across the perilous and busy shipping lane in recent months, with four deaths recorded in 2019 and now seven so far this year.
In September, French authorities said they had intercepted over 1,300 people trying to reach the UK, including a handful who had attempted to swim the 30-odd kilometres (18.6 miles) across the Channel.
Northern France has long been a magnet for people seeking to smuggle themselves to Britain in small boats or in one of the tens of thousands of trucks and cars that cross over daily on ferries and trains.
The issue has been a source of tension, with Britain accusing France of not doing enough to stop the crossings.
Home Secretary Priti Patel said in a tweet she was "truly saddened" by the deaths and that Britain has offered its support to France in investigating the incident.
"The tragic news highlights the dangers that come with crossing the channel and I will do everything I can to stop callous criminals exploiting vulnerable people," she added.
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