WASHINGTON — Tensions over President Trump’s nominations turned into a parliamentary game of chicken on Monday, with Republican leaders vowing to keep the Senate in session, day and night, until lawmakers confirm four of his cabinet picks.

After days of grasping at procedural hurdles, Democrats held vigil against Betsy DeVos, Mr. Trump’s polarizing nominee for education secretary, and promised to spend the final 24 hours before her confirmation vote Tuesday reiterating their objections.

But as Democrats made clear they had no intention of yielding even a minute of their allotted floor time to debate Mr. Trump’s nominations — a final act of parliamentary disobedience for a minority party that lacks the votes to block a nominee on its own — Republicans stood their ground.

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Daring Democrats to keep their word at the expense of several sleepless nights, Senator John Cornyn of Texas, the chamber’s No. 2 Republican, said the Senate session would not end until lawmakers confirmed four of Mr. Trump’s nominees: Ms. DeVos, Senator Jeff Sessions of Alabama as attorney general, Representative Tom Price of Georgia to lead the Department of Health and Human Services, and Steven T. Mnuchin as Treasury secretary.

The effort could last into Saturday and might involve burning “a little midnight oil,” Mr. Cornyn said.

Short one critical but elusive Republican “no” vote against Ms. DeVos — a billionaire philanthropist with little experience in public schools and a remarkably shaky showing at her confirmation hearing — Democrats on Monday did the only thing they could: They talked.

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Senator Patty Murray of Washington announced that Democrats would occupy the floor until the vote on Ms. DeVos’s nomination, which is expected around midday on Tuesday. Ms. Murray is the top Democrat on the committee that approved Ms. DeVos along a party-line vote, and one of her most strident opponents.

Publicly, Democrats held out hope that they could woo one more Republican dissenter to join Senators Susan Collins of Maine and Lisa Murkowski of Alaska in voting against Ms. DeVos. Both announced last week that they would not support Ms. DeVos’s nomination.

Their defections, combined with a unified vote from the Democratic caucus, set up a 50-50 tie on the nomination, which Vice President Mike Pence would have to break in his capacity as president of the Senate.

Appearing later at a demonstration outside the Capitol, Ms. Murray urged protesters to pressure Republicans to oppose Ms. DeVos. “If we can persuade one more Republican to do the right thing, we can double down on the message we’re all sending to President Trump: The Senate stands with public education!” Ms. Murray said.

A deluge of constituent calls and messages against Mr. Trump’s nominees and executive orders has overwhelmed some Senate offices and even the Capitol phone system in recent days, echoing demonstrations across the country. Energized by the opposition, Democrats seemed ready on Monday to resort to sleeping bags if necessary.

Senator Chris Van Hollen, Democrat of Maryland, said Ms. DeVos had embraced a philosophy that abandoned poorer families. She has been vehemently criticized by many education advocates for favoring voucher programs over repairing public schools.

“What do we say to them?” Mr. Van Hollen asked.