CENTREVILLE, Md. — The U.S. Department of Agriculture’s field office on Maryland’s Eastern Shore was closed because of the partial federal government shutdown when George Godfrey Jr. showed up Wednesday, leaving him without a reimbursement check for his grain crop losses.

Despite the setback, Mr. Godfrey didn’t waver in his support for President Trump. In fact, he said he was ready to hang tough indefinitely with Mr. Trump in the shutdown fight over border security.

“I pretty much agree with the guy 100 percent. I think his opposition is the problem,” said the 62-year-old farmer. “I’ve had some really difficult times, but there are too many people in the country that just want free handouts, and I think that’s where all this noise is coming from.”

Count Mr. Godfrey among the president’s base that remains rock-solid as the partial government shutdown drags on for a fourth week.

A Quinnipiac University poll released this week found that 86 percent of Republican voters approve of Mr. Trump’s performance, up from 82 percent in December before the shutdown.

Tim Malloy, assistant director of the poll, called the president’s base “granite strong.” He credited their loyalty with keeping Mr. Trump’s overall approval rating above 40 percent despite his poor showings on honesty, empathy, leadership and fitness to serve.

What’s more, Mr. Trump sustained the party’s support despite polls showing that Americans blame him more than Democrats for the partial government shutdown.

Standing firm in opposition, Democrats say Mr. Trump’s demand for $5.7 billion for a fence or barrier is an ineffective and wasteful border security strategy.

House Speaker Nancy Pelosi, California Democrat, calls the wall “immoral” and has signaled to the party’s liberal base that Mr. Trump’s plan is racist.

Republican Party strategist Ford O’Connell said the illegal immigration issue is what unites and energizes Mr. Trump’s supporters, which is why the crowds at campaign rallies in 2016 spontaneously broke out in chants of “Build the wall.”

“Most Republicans recognize that Democrats have no interest in getting a handle on illegal immigration. In fact, Democrats are doing their best to incentivize illegal immigration with the explosion of sanctuary cities across the country, driver’s licenses for illegals, free health care and even voting rights in some cases for illegal aliens,” Mr. O’Connell said.

“Republicans see the writing on the wall, and they recognize that Donald Trump is the last, best chance to get illegal immigration under control.”

Stephanie Guiles, a legal aide who lives in Easton, Maryland, said the fight against illegal immigration keeps her firmly ensconced in the Trump camp, though she worries about federal workers going without paychecks. She said her cousin is a Transportation Security Administration agent.

“I’m sick of illegal aliens. I know all about illegals, and I thank God I don’t live in California anymore,” said Ms. Guiles, 50. “They want our illegal aliens to have housing [and] health benefits, but our own veterans are not being taken care of.”

The question for the White House is how long the support will hold if the shutdown lasts several more weeks or even months.

White House economic adviser Kevin Hassett said Tuesday that the partial shutdown is delivering a bigger blow to the economy than he originally projected.

The president’s Bureau of Economic Advisers now calculates that lost wages of federal contractors and employees, coupled with their reduced spending and investment, will lower economic growth by 0.13 percent per week and that the hit will increase over time.

However, Mr. Hassett said the economy will rebound when the shutdown ends and the workers collect their back pay.

The administration has made several moves to soften the impact of the shutdown.

The Internal Revenue Service is calling back about 46,000 employees to work unpaid while processing income tax refunds, which will help reduce the drag on the economy.

Offering relief to people such as Mr. Godfrey, the USDA on Wednesday recalled about 2,500 furloughed employees to temporarily reopen some Farm Service Agency offices.

The offices will be open on Thursday, Friday and Tuesday during normal business hours, helping farmers and ranchers process loans and tax filings.

“We are doing our best to minimize the impact of the partial federal funding lapse on America’s agricultural producers,” said Agriculture Secretary Sonny Perdue.

Polling suggests that most Americans are not feeling much pain from the shutdown. According to an ABC News/Washington Post poll this weekend, 4 in 5 people said they haven’t been affected.

• Dave Boyer contributed to this report.

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