Frank Ospina, 39, came to the UK in late 2022 to visit his mother and took a short-term job washing dishes.
He was detained at Heathrow detention centre after an immigration raid in March where his mental health deteriorated and he died.
Speaking to the BBC, his family said he repeatedly asked for help and was not seeking to avoid deportation.
His sister, Tatiana Rios Ospina, told the broadcaster: “My brother was a civil engineer who graduated from the best public university in Colombia.
“He was hard-working and very intelligent. They took him to the limit of sanity.”
Mr Ospina’s brother-in-law, Julian Llano, added: “He kept insisting that he felt very bad, mentally, that he needed to get out of there.
“He didn’t ask for help - he begged for help, not only to us, but also to the people there."
The exact circumstances of Mr Ospina’s death on March 26 will be established by a coroner.
According to documents obtained by Liberty Investigates, an “attempted mass suicide” then took place at the centre in protest at conditions days after Mr Ospina’s death.
Files obtained by the outlet state that between four and six detainees tried to kill themselves during the protest. Home Office contractor Mitie said custody officers managed to stop the attempts.
A Home Office spokesman said: “Our thoughts and condolences are with the family and friends of Mr Ospina.
“The welfare of all those in our care is of the utmost importance. Any death in immigration detention is a tragic event and will be subject to investigation by the police, the coroner, and the independent Prisons and Probation Ombudsman.”
The latest case at Harmondsworth IRC will heap further pressure on the Home Office over conditions at the centres, which are prison-like facilities for asylum seekers or other migrants to be detained until deportation or their status is otherwise resolved.
The UK has no maximum limit on the length of time someone can be detained, with the sole exception of pregnant women.
A public inquiry into Brook House, an IRC near to Gatwick, is due to report next week after a BBC expose revealed alleged instances of abuse and mistreatment of detainees.
An inspection report last year said the facility did not meet the mental health needs of detainees but Home Office officials said work was ongoing to address those concerns.
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