The early-morning ruling from a federal appeals court was the latest chapter in a saga which began on January 27.
The early-morning ruling from a federal appeals court was the latest chapter in a saga which began on January 27, when Trump issued a blanket ban on all refugees, and travelers from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.
For now, the decision maintains one made by a federal judge in a lower court, who temporarily suspended Trump's order on Friday pending a wider legal review.
The next deadline comes on Monday, when all parties must submit additional documents to the appellate judges, according to a schedule determined by the court.
Trump initially dispatched Vice President Mike Pence to convey the White House's position on Sunday's political talk shows. Pence called the decision "frustrating."
"We will move very quickly," he told Fox News. "We are going to win the arguments because we will take the steps necessary to protect the country, which the president of the United States has the authority to do."
But in the mid-afternoon, after taking an uncharacteristic, nearly day-long break from Twitter, Trump came out swinging again.
"Just cannot believe a judge would put our country in such peril. If something happens blame him and court system. People pouring in. Bad!" he wrote.
"I have instructed Homeland Security to check people coming into our country VERY CAREFULLY. The courts are making the job very difficult!"
Trump already had unleashed a string of fiery tweets on Saturday defending his policy and attacking "so-called" federal judge James Robart, who issued Friday's decision in Seattle.
Asked by multiple networks whether Trump's comment about Robart was out of line, Pence defended his boss.
"Every president has a right to be critical of the other branches of the federal government," Pence told CBS News.
But Republican Senate majority leader Mitch McConnell told CNN: "I think it is best not to single out judges for criticism."
"We all get disappointed from time to time at the outcome in courts on things that we care about," he said.
Trump's executive order slapped a blanket ban on entry for nationals of the seven countries for 90 days and barred all refugees for 120 days. Refugees from Syria were blocked indefinitely.
In its appeal to Robart's decision filed late Saturday, the Justice Department said suspending the ban was causing "irreparable harm" to the American public.
It said Robart's ruling had run afoul of constitutional separation of powers, and "second-guesses the president's national security judgment."
But the Ninth Circuit Court of Appeals rejected the request for the travel ban to be immediately reinstated, without offering a reason.
The court asked the states of Washington and Minnesota, which had filed the original suit over the ban, to provide additional documents by 0759 GMT Monday.
And the Justice Department was given until 2300 GMT Monday to complete its legal dossier.
Then, the court could schedule a hearing, or rule on whether the ban should remain suspended.
Meanwhile, travelers from the targeted countries with valid visas began arriving on American soil.
In New York, 33-year-old Sudanese doctor Kamal Fadlalla rejoiced -- after a week blocked in his home country, he was back in the Big Apple with friends and colleagues.
"It feels great," Fadlalla told AFP on Sunday at John F. Kennedy International Airport. "It was a tough week actually."
Iranian graduate student Sara Yarjani, who was initially deported under Trump's order, arrived in Los Angeles.
"I am so grateful to all the lawyers and others that helped me," she said tearfully.
In Syria, a 25-year-old law graduate who asked not to be named said he was driving to Beirut on Sunday to catch a flight to Amman and then a connecting flight to New York.
"I jumped up and haven't been able to sleep since. I'm ecstatic," the man told AFP.
The department had earlier said up to 60,000 people had their visas revoked as a result of Trump's order.
The restrictions fueled numerous weekend protests at home and abroad -- from London and Hong Kong to Washington and Palm Beach, where Trump spent the weekend at his Mar-a-Lago retreat on Florida's east coast.
On Sunday, dozens of Trump supporters rallied in front of Trump Tower in New York, urging Americans to give him a chance -- but they ended up in a face-off with a dozen counter-protesters.