Police were called after the people were found on the container, which arrived on a ship from Zeebrugge in Belgium and was being unloaded at 7.35am by Port of Tilbury authorities.
Superintendent Roe said that crews were alerted to the people inside when they heard "kicking and screaming". Describing them as victims of "people trafficking", Mr Roe said they had been in the container a "significant amount of time" and that now police were working with international agencies to establish their movements prior to arriving in the UK.
The East of England Ambulance Service NHS Trust said in a statement: "We confirm that 35 patients were involved in the incident at Tilbury Docks, Essex; one patient has sadly been declared deceased at the scene, with others suffering from severe dehydration and hypothermia.
"After initial treatment by ambulance crews, all patients have now been conveyed to surrounding hospitals for further care; seven have been taken to Southend Hospital, nine to London Whitechapel Hospital and 18 to Basildon Hospital."
Natalie Hardy, from P&O Ferries, said the immigrants were in a container on board the Norstream, a commercial vessel which carries freight between Zeebrugge and Tilbury.
Ms Hardy said: "They (port authorities) found 35 on a container in the ferry. They had been in there overnight, because the ship was an overnight freight ferry.
"This morning when they went to unload containers there was a noise heard, a banging. One unfortunately was dead and others were taken to hospital.
"UK Border Agency and all emergency services were called and went to the scene and have been handling it since."
Public Health England said it is not involved at this point and had not been notified of there being any Ebola risk.
The spokeswoman added: "If it was Ebola, health care professionals are so alert at the moment to signs and symptoms that should there have been anyone who was showing symptoms we would have been notified immediately.
"I think we can be confident that we are not dealing with that."
A spokesman for Darent Valley Hospital in Dartford, Kent, said it was "currently on standby to receive cases".
Police are dealing with the incident alongside staff from Port of Tilbury, UK Border Force and the East of England Ambulance Service, the spokesman said, while a casualty bureau is being set up.