Rhinelander is closer to the Arctic Circle than to Mexico, so it is no great surprise that few people in the small Wisconsin city have laid eyes on the foreign migrants Donald Trump claims are “invading” the country from across the US border 1,500 miles to the south.

But Jim Schuh, the manager of a local bakery, is nonetheless sure they are a major problem and he’s voting accordingly.

“We don’t see immigrants here but I have relatives all over the country and they see them,” he said. “That’s Biden. He’s responsible.”

Large numbers of voters in key swing states agree with Schuh, even in places where migrants are hard to find as they eye cities such as Chicago and New York struggling to cope with tens of thousands of refugees and other arrivals transported there by the governors of Texas and Florida.

Trump has been pushing fears over record levels of migration hard in Wisconsin where the past two presidential elections have been decided by a margin of less than 1% of the vote. A Marquette law school poll last month found that two-thirds of Wisconsin voters agree that “the Biden administration’s border policies have created a crisis of uncontrolled illegal migration into the country”.

Trump has twice held rallies in Wisconsin over the past month at which migrants have been a primary target. In Green Bay he called the issue “bigger than a war” and invoked the situation in Whitewater, a small city of about 15,000 residents in the south of the state.

Republican politicians have turned Whitewater into the poster child for anti-migrant rhetoric in Wisconsin after the city’s police chief, Dan Meyer, appealed for federal assistance to cope with the arrival of nearly 1,000 people from Nicaragua and Venezuela over the past two years.

Meyer made clear in a letter to President Joe Biden in December that he was not hostile to the foreign arrivals as he expressed concern about the “terrible living conditions” endured by some.

“We’ve seen a family living in a 10ft x 10ft shed in minus 10 degree temperatures,” he wrote.

But the police chief said that his department was struggling to cope with the number of Spanish-speaking migrants because of the cost of translation software and the time taken dealing with a sharp increase in unlicensed drivers. Meyer also said that his officers had responded to serious incidents linked to the arrivals including the death of an infant, sexual assaults and a kidnapping.

However, he told Biden that “none of this information is shared as a means of denigrating or vilifying this group of people … In fact, we see a great value in the increasing diversity that this group brings to our community.”

That did not stop Republican politicians from descending on Whitewater to whip up fear.

The Wisconsin senator Ron Johnson, a close ally of Trump who has spoken at the former president’s political rallies, and a Republican member of Congress from the state, Bryan Steil, held a meeting in the city to denounce what they described as the “devastating” consequences of the migrant arrivals.

Johnson blamed “the whole issue of the flood of illegal immigrants that have come to this country under the Biden administration”.

Steil declined to back Meyer’s appeal for federal financial assistance and said the answer lay in legislation to secure the border. However, the congressman was among those Republicans who killed off a bipartisan border security law after Trump opposed the legislation in an apparent move to keep the crisis a live political issue going into the presidential election.

Republican members of the Wisconsin legislature wrote to Biden in January demanding action over what they claimed was a surge in violent crime in Whitewater even though Meyer has said he sees no threat to residents from the migrants and that “we are a safe community”.

Some Whitewater residents are furious at the political intervention. Brienne Brown, a member of the city council for six years, said residents had been welcoming of the migrants, with community organisations providing food, furniture and bedding to many.

“The spotlight fell on us because Ron Johnson and Bryan Steil decided to make it a political event for themselves. Most people here were incredibly angry. They feel like they’ve been used as a political football,” she said.

“The crime that is occurring is super low level, which is mostly our police department pulling over somebody in a car who doesn’t have a licence.”

The police chief has called for migrants to be allowed to obtain driving licences but the Wisconsin legislature will not allow it.

Brown said that the serious incidents of assault involved domestic violence as well as the case of a woman who abandoned her newborn baby in a field, and that those kind of crimes remained uncommon.

Wisconsin has long relied on migrant workers, many of them undocumented, as farm labour. Studies have suggested that the state’s dairy farms would grind to a halt without foreign workers. Historically, most were from Mexico. Whitewater tended to attract people from Guanajuato as migrants from the Mexican state sent word back about job opportunities.

Brown noticed a change during the Covid crisis.

“I’d knock on doors a lot just to talk to my constituents right around the pandemic. I started noticing that a lot of them were not from Mexico. They were from Nicaragua and Venezuela,” she said.

Brown said the workers moved into accommodation left by students forced to return home by the pandemic lockdown.

“We have a lot of farms, a lot of chicken farms, a lot of egg farms. There are factories that make spices, there are factories that can food. They’re always looking for low-paid workers and they never have enough. So there was plenty of work available,” she said.

Schuh, like many other Americans critical of what they describe as Biden’s open border policy, makes a point of distinguishing between those who go through the formal process of immigration with a visa and those walking across the border to seek asylum or work illegally.

“I have nothing against immigrants but it has to be done the right way,” he said.

Trump continued to stoke the issue at a rally in Michigan earlier this month when he blamed Biden for the murder of Ruby Garcia in March. The former president claimed his administration had deported the man who has confessed to the shooting, Brandon Ortiz-Vite, and that “crooked Joe Biden took him back and let him back in and let him stay in and he viciously killed Ruby”. Ortiz-Vite was deported in 2020 following his arrest for drinking and driving. It is not clear when he returned to the US.

Trump told the rally that he spoke to Garcia’s family and that they were “grieving for this incredible young woman”. But Garcia’s sister, Mavi, denied that anyone in the family spoke to the former president and accused him of exploiting the murder for political ends.

“He did not speak with any of us, so it was kind of shocking seeing that he had said that he had spoke with us, and misinforming people on live TV,” she told WOOD-TV.

“It’s always been about illegal immigrants. Nobody really speaks about when Americans do heinous crimes, and it’s kind of shocking why he would just bring up illegals. What about Americans who do heinous crimes like that?”