They were discovered yesterday after a freighter arrived at Tilbury Docks in Essex from Belgium.
The 35 people found inside the container are understood to be Sikhs from Afghanistan, Essex Police said.
Earlier, the police described them as people believed to be from the Indian sub-continent and questioned some of them.
A homicide investigation is continuing into the death of one of the 35 rescued.
They were treated for severe dehydration and hypothermia before being taken to a detention centre.
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Essex Police described the men, women and children as victims of "people trafficking" and are working with Interpol and other international authorities to try to establish what happened.
Many of the survivors spent the night in hospital but none is thought to have suffered serious injury.
One man is believed to have been questioned by officers already and most of the others have been discharged from hospital.
The Indian Ministry of External Affairs (MEA) has confirmed being in contact with UK authorities to ascertain their nationality.
"It is a homicide investigation...We will be looking to see where the origin and the gangs or whoever may [be] involved in this conspiracy to bring these people in this way over to this country. Clearly we need to try and bring them to justice," Supt Trevor Roe of Essex Police told reporters at a press conference.