Video report by ITV News Correspondent Dan Rivers
Her family says Ms My has a distinctive marking on her foot and are appealing for information from police with regard to her fate. Officers have not confirmed the identity of any people found dead in the back of the lorry.
In a series of short messages, she says:
"I'm sorry mum.
"My path abroad hasn’t succeeded.
"Mum, I love you so much! I'm dying because I can't breathe.
"I'm from Nghen, Can Loc, Ha Tinh. Vietnam.
"I am sorry, Mum."
The Vietnamese Embassy said staff will arrange a visit to site in a bid to identify victims after Vietnamese families said their loved ones may be among the deceased.
It continued: "The Embassy has not yet received any official confirmation from the British relevant agencies. The Embassy has continued to follow the event and worked closely with relevant British agencies to accelerate the investigation process."
As well as the lorry's driver, three further people have been arrested in connection with the deaths.
Mo Robinson, the Northern Irish lorry driver detained in the early hours of Wednesday morning, remains in police custody.
A 48-year-old man from Northern Ireland has been arrested on suspicion of conspiracy to traffic people and suspicion of manslaughter. Essex Police said the man was arrested at London Stansted Airport.
A man and a woman, both 38-years-old, were arrested in early morning raids in Warrington, Cheshire on Friday morning.
A statement from Essex Police said: "We have carried out warrants in Cheshire as part of the investigation into 39 bodies being discovered in a lorry trailer in Grays .
"As a result, a 38-year-old man and a 38-year-old woman from Warrington have been arrested on suspicion of conspiracy to traffic people and on suspicion of 39 counts of manslaughter.
"A 25-year-old man, the driver of the lorry, remains in custody on suspicion of murder. A warrant of further detention was granted yesterday, Thursday 24 October, by local magistrates."
Police have appealed for people who believe they may have been trafficked to come forward to assist with the investigation.