A teenager is confirmed to be among the four people who died, while four others are still missing.
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A rescue operation in the English Channel for up to four more people missing at sea is continuing.
On Wednesday, a packed migrant boat started sinking in freezing waters between Kent and France.
Crews on a nearby fishing boat and lifeboats rescued 39 people, but four people were confirmed dead. The latest search will be carried out by drone.
In a joint statement, the UK and France pledged to “destroy” the business model of people-smuggling gangs.
Home Secretary Suella Braverman and her counterpart Gérald Darmanin said the tragedy highlighted the need to jointly prevent the crossings.
“Our hearts go out to all those affected by this tragic event,” they said.
The Maritime and and Coastguard Agency said some searches had taken place overnight with vessels in the area asked to post lookouts and report sightings to the Dover Coastguard.
Footage from Wednesday’s rescue showed the inflatable boat filling with water as some dressed only in T-shirts and thin lifejackets screamed for help.
The video – shared by the owner of the fishing trawler, Ben Squire – showed crew pulling people up out of the water and the boat with ropes.
Those rescued from the dinghy said they had each paid £5,000 to cross the channel, the BBC has been told.
After hauling people to safety, Mr Squire said the crew gave them hot showers, their own clothes and fed them to help warm them up.
Charles Blyth, the safety officer at the company which owns the trawler, said it was “sheer coincidence” it was in the right place for the rescue.
“As soon as the individuals on that dinghy saw [our] fishing vessel, many of them started to jump off and swim,” he told BBC Radio 4’s Today programme.
He said crew members onboard had been trained in emergency care for people going overboard.
Temperatures overnight dropped below 1C and were likely to have been colder out at sea.
On Wednesday, Prime Minister Rishi Sunak expressed his sorrow at the “tragic loss of human life”. The Archbishop of Canterbury Justin Welby said debates about asylum seekers were not about statistics but “precious human lives”.
A French organisation, Utopia 56, which helps migrants in Calais, said it was contacted at 01:53 GMT – with a voice message and a location – by a boat in distress in the Channel.
Nikolai Posner from the organisation said the voice message stated there were people in the water and families on board.
“It was clearly an emergency, he was calling for help,” he told PA News, adding that babies could be heard “screaming” in the background.
The organisation, however, has said it was not possible to verify whether the distress call was definitely from the boat in question.
In November 2021, at least 27 migrants died after a dinghy sank while heading to the UK from France.
Some 460 people made the journey from France to Kent in small boats between Friday and Sunday, the BBC’s Simon Jones said.
Nearly 45,000 people have made the journey this year so far.
