Early indications suggest 38 are adults and one aged in their teens, police said.
Essex Police said it was called by the ambulance service shortly before 1.40am on Wednesday following the discovery of a lorry container with people inside at Waterglade Industrial Park, Grays
Chief Superintendent Andrew Mariner said: "This is a tragic incident where a large number of people have lost their lives. Our enquiries are ongoing to establish what has happened.
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"We are in the process of identifying the victims, however I anticipate that this could be a lengthy process."
"We believe the lorry is from Bulgaria and entered the country at Holyhead on Saturday 19 October and we are working closely with our partners to investigate."
"We have arrested the lorry driver in connection with the incident who remains in police custody as our enquiries continue."
Home Secretary Priti Patel tweeted: "Shocked & saddened by this utterly tragic incident in Grays. Essex Police has arrested an individual and we must give them the space to conduct their investigations."
A restaurant worker in the industrial estate, who did not want to be named, said the area was cordoned off by police on Wednesday morning. "There's a lot of police and forensics"
Richard Burnett, chief executive of the Road Haulage Society, said "this tragedy it highlights the danger of migrant gangs people-smuggling on lorries".
Independent MP Nick Boles tweeted: "Every now and then something truly appalling breaks through the Brexit fog enveloping Parliament and reminds us how many more important issues are being neglected"
"I am receiving regular updates and the Home Office will work closely with Essex Police as we establish exactly what has happened."
Prime Minister Boris Johnson tweeted: "I'm appalled by this tragic incident in Essex."
"My thoughts are with all those who lost their lives & their loved ones."
Conservative MP for Thurrock Jackie Doyle-Price wrote on Twitter: "Sickening news of 39 people found dead in a container in Grays."
"People trafficking is a vile and dangerous business. This is a big investigation for Essex Police. Let's hope they bring these murderers to justice."
It is thought the lorry will have entered Holyhead via Dublin.
Seamus Leheny, Northern Ireland policy manager for the Freight Transport Association (FTA), told PA: "There is a direct route to Holyhead from Dublin."
"If the lorry came from Bulgaria, getting into Britain via Holyhead is an unorthodox route."
At a press conference at Grays police station, deputy chief constable Pippa Mills said police were unable to identify the bodies.
She said: "Please appreciate we are in the early stages of what is likely to be a lengthy investigation."