"This is the heaviest toll we've ever had in the north," Herve Tourmente, the top state official in Dunkirk, told French radio France Bleu Nord.
A man, a woman, and two children aged 8 and 5 were found dead, according to what the prefecture said was still a provisional toll. Another 15 people were rescued, with eight suffering from hypothermia and two in cardiac arrest, local firefighters said earlier on Tuesday.
A major search operation was launched after the boat capsized in the morning. Emergency and medical crews attended the scene, along with a French Navy helicopter.
"My thoughts are with the loved ones of those who tragically lost their lives in the Channel today," British Prime Minister Boris Johnson said on Twitter.
"We have offered the French authorities every support as they investigate this terrible incident and will do all we can to crack down on the ruthless criminal gangs who prey on vulnerable people by facilitating these dangerous journeys."
Several thousand migrants have attempted the dangerous crossing of the English Channel this year, often paying human smugglers to help them through one of the world's busiest shipping lanes in overloaded rubber dinghies.
Britain has repeatedly pressed President Emmanuel Macron's government to do more to prevent the migrants leaving France.
