The woman, Joy Gardener, aged about 40, is understood to have stopped breathing after a struggle with police officers at her home in Hornsey, north London. She then collapsed.
She was given mouth-to-mouth resuscitation and taken to Whittington hospital, where her condition yesterday was described as critical.
William McCall, of the PCA, will supervise the investigation by Detective Superintendent Kevin Proctor of the Metropolitan Police.
Mr McCall said: 'I will ensure that a full and impartial investigation is carried out into this sad incident. I intend to find out exactly what happened. There will be no cover-up.'
The Home Office will also examine the case after the immigration minister, Charles Wardle, called for a full report. A Home Office spokesman said: 'There is scope to reconsider the case in the face of fresh compassionate circumstances.'
He said the woman had arrived in Britain on a six-month visitor's visa and had come to the attention of immigration officials when she applied for an extension of her stay in late 1990.
She was told to leave Britain, an appeal was turned down and police and immigration officials were serving a deportation order when the incident happened.
The Home Office said the woman's five-year-old son was born in Britain, but this did not give him the right to British citizenship. Although the deportation order named only the woman, both would be expected to leave. The child, who was present during the incident, is in the care of social services.
The woman had moved to Hornsey from nearby Tottenham just a few weeks ago. Before her collapse, her case had been taken up by Tottenham's Labour MP, Bernie Grant, who has called for an independent inquiry.