The barrister representing a British family in their bid to stay in Australia says they have been given a short extension to appeal to Immigration Minister Peter Dutton.

The Immigration Department has rejected a residency application by six-year-old Sienna Tippett, who currently lives at Lennox Head on the New South Wales far north coast.

Her parents moved to Australia from the United Kingdom more than five years ago, when Sienna was 10 months old, and both have been granted Australian residency.

They said their daughter had an undiagnosed medical condition affecting her balance and speech.

Barrister Stephen Lawrence said Sienna would be forced to leave the country unless Mr Dutton intervened.

"There has been a groundswell of support from the community and the Tippetts have gathered a lot of what, I think, is impressive medical and community evidence that will be used in support of their application," he said.

"There really are some special aspects to this case that we will be asking the minister to consider.

"It's a discretion that has to be exercised by him personally and a discretion in which he is allowed to take into account a range of factors and effectively it's very difficult to challenge those sort of absolute discretions.

"So it's very much over to him to consider the broad range of circumstances and make a decision that sees this family able to stay in Australia."

The minister is making no comment at this stage.