A US appeals court has rejected the Trump administration's request to immediately reinstate its travel ban barring citizens from seven mainly Muslim countries and temporarily banning refugees.

Key points: The US Justice Department filed a formal appeal on Sunday

The US Justice Department filed a formal appeal on Sunday The White House has until Monday to present a reply

The White House has until Monday to present a reply Who is the judge who blocked the ruling?

A federal judge temporarily invalidated the ban on Friday night (local time), and the US Justice Department on Sunday filed a formal appeal against the ruling.

"We'll win. For the safety of our country, we'll win," Mr Trump told reporters at his private Mar-a-Lago resort in Florida.

But on Sunday evening, the US Court of Appeals for the 9th Circuit denied the request.

"Appellants' request for an immediate administrative stay pending full consideration of the emergency motion for a stay pending appeal is denied," the ruling said.

The Trump administration had said Seattle Judge James L Robart overstepped his authority by temporarily blocking the ban nationwide.

The appeal court's denial of an immediate stay means the legal battles will continue for days at least.

A reply from the Justice Department is due on Monday.

"We'll accomplish the stay and will win the case on the merits," Vice-President Mike Pence said.

Members of Mr Trump's Republican Party scolded him for attacking Judge Robart on Twitter, and politicians accused the President of stepping over the line that separates the executive from the judiciary.

Mr Trump said Robart was a "so-called judge" whose "ridiculous" ruling would be overturned.

"The president can criticise anybody he wants," Mr Pence said.

"[The American people] find it very refreshing that they not only understand this president's mind, but they understand how he feels about things."

'The President is not a dictator'

Acting solicitor-general Noel Francisco forcefully argued on Saturday night (local time) that the President alone had the power to decide who could enter or stay in the United States — an assertion that invokes the wider battle to come over illegal immigration.

"The power to expel or exclude aliens is a fundamental sovereign attribute, delegated by Congress to the executive branch of government and largely immune from judicial control," a brief said.

The order has caused unending confusion for many foreigners trying to reach the United States, prompted protests across the country and led to multiple court challenges.

Senator Dianne Feinstein of the District of California also defended Mr Trump's actions.

"The President is not a dictator," she said.

"He is the chief executive of our country. And there is a tension between the branches of government."

Demonstrations took place outside the White House, in New York and near Mr Trump's estate in Palm Beach, Florida, on Sunday.

The ban sparked protests around America, including this one in West Palm Beach. ( AP: Michael Ares )

The State Department, after initially saying that as many as 60,000 foreigners from Iraq, Syria, Iran, Sudan, Libya, Somalia or Yemen had their visas cancelled, later reversed course and said they could travel to the US if they had a valid visa.

The Homeland Security Department is no longer directing airlines to prevent visa-holders affected by Mr Trump's order from boarding US-bound planes.

A spokesperson for the Iraqi Government expressed satisfaction with the decision.

"It is a move in the right direction to solve the problems that it caused," Saad al-Hadithi said.

A family embrace before boarding a flight from Iraq to the US. ( Reuters: Ahmed Saad )

Reuters/AP