Donald Trump’s immigration crackdown has a problem: not enough Americans are willing to carry it out.

The Border Patrol is losing agents faster than it can replace them, putting a question mark over the president’s plan to ramp up the force.

Air and Marine Operations, a separate agency, is also struggling to find pilots and other employees.

“If you know people who are enthusiastic about border security please send them to Customs and Border Protection (CBP),” Ronald Vitiello, the Border Patrol chief, said in an appeal this week. “We’re already behind. We’re not hiring fast enough to keep up with the attrition.”

He spoke at a border security expo in San Antonio, Texas, where other senior officials spoke out about recruitment woes.

Benjamin Huffman, head of the Border Patrol’s strategic planning and analysis, half-joked to an audience that everyone would have to submit five names of potential recruits before leaving.

Trump has ordered the agency to add 5,000 agents to beef up patrols and surveillance in advance of his proposed border wall. But its current 19,000-strong force is already 2,000 shy of a target set during the Obama administration.

Officials said tough screening, especially a lie-detector test, rejected many qualified candidates, and that tough conditions such as living in remote, rugged areas prompted more than 1,000 agents to quit every year.

“Some people just don’t want to live there,” said Randolph “Tex” Alles, acting deputy commissioner of CBP, a 60,000-strong agency that includes Border Patrol. “Hiring challenges are not new. Attracting and recruiting high quality individuals is a challenge for us.”

An agent mans a recruiting station at the Border Security Expo in San Antonio, Texas. Photograph: John Moore/Getty Images

Border Patrol officials are especially nervous that a planned expansion of Immigration and Customs Enforcement (Ice) – Trump has ordered it to add 10,000 agents – will precipitate a stampede to the sister organisation. “There’s a real concern that a lot of that will come from Border Patrol,” Huffman said.

Vitiello, his boss, was even bleaker: “[They] could get them all from CBP.”

Ice looks for undocumented people in the US, so its agents live in cities, not desert outposts, and the agency offers more overtime opportunities.

It has another recruitment advantage: no lie-detector test. About two-thirds of CBP applicants fail the polygraph, the Associated Press reported in January.

A 2010 law obliged CBP to use the test to curb corruption and misconduct after an earlier hiring surge doubled the Border Patrol’s size in eight years. The agency is lobbying Congress to let it exempt military veterans and candidates from other law enforcement agencies.

Alles said the change would not sacrifice quality for quantity. “It won’t serve us well to drop our standards. No law enforcement agency has ever grown rapidly and not had problems with corruption.”

But James Tomsheck, who was the agency’s internal affairs chief from 2006 to 2014, told the Associated Press last month that curbing the polygraph would degrade vetting and compromise integrity.

The economic recovery has created another headache for law enforcement recruitment.

Tony Crowder, the executive director of Air and Marine Operations, told the Guardian his agency was struggling to retain and recruit enough pilots.

Commercial airlines were luring pilots who in some cases were expected to work in remote areas and participate in arrests. “They pay more and it’s a different type of work.”

Another problem is the attitude of young people, especially those for whom 9/11 is distant history. “They have a different view of public service. I don’t want to indict an entire generation but it’s harder to sell self-sacrifice for the common good.”

Even so, Crowder said, morale at Air and Marine Operations was high.

CBP officials echoed that view, saying the administration’s focus on border security had piled pressure on but also energised the sense of mission.

The Border Patrol has accelerated application processing and revamped academy training, said Vitiello, the agency chief. “If you know people who are enthusiastic about border security please send them to CBP,” he told an audience. “It’s awesome.”