More than a week after the death in custody of a disabled Warlpiri man in Mparntwe/Alice Springs, no Northern Territory police officer involved in his arrest has been stood down, despite an investigation taking place.

Kumanjayi White, a 24-year-old Warlpiri man living in supported accommodation and under state guardianship, died last week after being restrained by two plain-clothed NT police officers following an alleged altercation with a security guard at a Coles Supermarket.

The NT Police and Chief Minister Lia Finocchiaro have rejected calls from Mr White's family, human rights groups, and Indigenous Australians Minister Malarndirri McCarthy for an independent investigation into the death — despite long-standing concerns from Indigenous communities about systemic racism and over-policing in the Territory.

It was reported last week that one of the officers allegedly involved in the initial arrest of Mr White was Mparntwe/Alice Springs police prosecutor Steven Haig. When asked to confirm his involvement, NT Police issued a conflicting response.

While confirming the death was being investigated by the Major Crime Division — which NT Police said "operates under strict protocols and with full transparency" — they also told National Indigenous Times they would "not be identifying the officers involved in the arrest".

After subsequently inquiring if the officers allegedly involved in Mr White's arrest were still actively on duty, NT Police replied: "The NT Police can confirm no one has been stood down."

This means the officers allegedly involved in Mr White's death — allegedly including a prosecutor — are continuing to serve in Mparntwe/Alice Springs, a town facing an incarceration crisis, particularly among First Nations people, despite an investigation into the event taking place.

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Mr White's grandfather, Warlpiri Elder Ned Jampijinpa Hargraves, told the ABC on Tuesday the family was "very, very angry" with the police response.

"When we met at the police station with the coroner and with the commissioner [it was] the same story that we hear every day every time, that this has happened and the police did their job," he said.

"We are sick of it and we are losing lots of our loved ones … we want to get the police to be accountable because it just keeps going."

Despite stating their commitment to transparency, NT Police have also refused to release CCTV footage of the incident. Mr Hargraves told the ABC the family wants the footage and an independent investigation "so we can get a better understanding of what happened".

Independent Senator Lidia Thorpe said this week the NT Police's handling of the case has been "disrespectful and predictable" accusing the force of "spinning a narrative that blames Kumanjayi" while simultaneously refusing to release the CCTC footage which could "help the family better understand his death".

Acting Police Commissioner Martin Dole has defended the internal investigation, arguing an independent inquiry could compromise "further action being taken" — despite oversight bodies in other states, such as the Law Enforcement Conduct Commission in NSW, routinely handling such investigations.

Chief Minister Lia Finocchiaro has also categorically ruled out an independent inquiry. On Tuesday, she said it was "entirely appropriate" for police to investigate themselves and described the process as standard across the country.

"People can have full confidence in the Northern Territory Police Force to do their job, which is investigate matters … this now needs to take its course, and I urge everyone to respect the process," she told ABC Radio Darwin.

When asked about comments by Senator McCarthy — a widely respected Aboriginal woman from the NT — the Chief Minister said she had spoken to the Minister and found her remarks "really unhelpful".

"If she wants to support the people of Yuendumu and people concerned about this, then the best thing she can do is use her powerful voice to call for calm and confidence in the NT Police Force," she said.

Greg Barns SC, national criminal justice spokesperson for the Australian Lawyers Alliance, said an independent investigation should not only "scrutinise the police conduct but should also extend to the Northern Territory Government who had a duty of care for this man".

"The investigation into this death needs independent oversight," Mr Barns said.

"Unfortunately, police cannot be trusted to investigate their own, particularly in a small jurisdiction, and neither can the NT Government be trusted to investigate its duty of care failure. An independent investigation will ensure transparency and fairness for all involved, including the police."