The Tamil mother who was due to be deported from Australia with her husband and two Australian-born children says a temporary extension preventing her daughter's removal gives them “some relief” and they remain hopeful of a permanent stay.

On Friday, a Federal Court judge granted an extended injunction preventing the deportation of two-year-old Tharunicca from Darwin to Sri Lanka until after another court hearing next Wednesday.

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While it was not immediately clear what the ruling meant for the rest of the family, Carina Ford - the lawyer for Priya, her husband Nadesalingam and their two Australian-born daughters - said it would be "not a good look" to deport the rest of the family and leave the child here alone.

"The injunction covers the two-year-old child. And obviously it will be left to the department and the minister to determine what to do with the remaining members of the family," she said.

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"So the fundamental argument that we're running in the Federal Court at the minute is that the youngest child was never part of the applications to the High Court or for any applications that have previously been run.

"And it's our view that that child needs their claims to be considered independently."

The lawyer hopes that by halting deportation for the infant, the rest of the family will also avoid deportation.

"I think it's hard to argue that her being removed from her parents is in her best interest. I mean, she's two years' old," she said.

"She's already had a traumatic experience in detention, which has been well covered by the media.

"I think the argument is that clearly having her parents with her, even if it is that if she remains in immigration detention, which we're not supportive of, it's beneficial for her to have her parents with her."

The family had spent Friday night in a Darwin hotel, and Priya told SBS Tamil she was relieved to hear the news of the reprieve.

On Friday morning, the Federal Circuit Court decided to grant her youngest daughter a temporary extension preventing her deportation until Wednesday.

Tamil family given last minute reprieve injunction ordered until next Wednesday @SBSNews — Gloria Kalache (@gloriakalache) August 30, 2019

Supporters for Tamil family singing the girls favourite song twinkle twinkle little star outside the Federal Court ⁦@SBSNews⁩ pic.twitter.com/hCUV3V7msD — Gloria Kalache (@gloriakalache) August 29, 2019

Group of supporters for the Tamil family set to be deported outside the Federal Court in Melbourne ⁦@SBSNews⁩ pic.twitter.com/icC52nzC89 — Gloria Kalache (@gloriakalache) August 29, 2019

“This decision gives us some relief, I don’t know what else to say,” Priya told SBS Tamil on Friday morning, shortly after the hearing in Melbourne.

Priya said she remains hopeful her family will be able to stay on, thanking supporters who had been rallying against the family’s deportation.

“The people of Australia are very kind, they have humanitarian values. They will accept me as a refugee,” she said.

“They have showered our family with love for the past 17 months. I am very hopeful that their love will succeed one day soon.

Priya pleaded to Home Affairs Minister Peter Dutton to exercise ministerial intervention in her family's case.

"Our children were born here. They don’t know any life outside here. They have gone through a lot. They can’t face any more trauma," she said.

"I plead with the Home Affairs Minister to take another look at our situation, open his heart and grant us a safe life here – for the sake of our children. That is my hope."

But Mr Peter Dutton said the family were not refugees.

"They're not owed protection by our country," he told Nine's Today program on Friday.

"They came here by boat and we've been very clear that they wouldn't stay."

Last week, the government blocked an application for an assessment of the dangers that toddler Tharunicaa may face if she is sent to Sri Lanka.

That move was effectively the family's last avenue to staying in Australia.

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Priya and Nadesalingam came to Australia separately by boat in 2012 and 2013, following Sri Lanka's civil war. They settled in Biloela for four years on a temporary bridging visa, which ran out in March 2018.

The High Court denied their bid to stay in May.

The family claims they would face persecution in Sri Lanka because of past family links to the banned Liberation Tigers of Tamil Eelam.

Last week, the Department of Home Affairs told SBS News "this family's case has been assessed, over many years, by the department and various merits review bodies. These decisions have also been the subject of judicial review applications in the courts".

Emergency: deportation attempt- supporters please go to Melbourne Airport and support this family with peaceful action now. Please share. pic.twitter.com/YTG8Agxxur — HometoBilo (@HometoBilo) August 29, 2019

Sri Lankan Consul General to Australia Lal Raj Wickrematunga has previously told SBS News it was "safe" for the family to return.

"As far as the Sri Lankan government is concerned, Sri Lanka is safe for Tamil families to return.

"The government has made an appeal for all those who've left Sri Lanka and sought refugee status elsewhere to come back."

With additional reporting from AAP