In the vast Alpine factory where Swiss Army knives are made, workers fixed finely sharpened blades onto stainless steel plates. Corkscrews, mini-scissors and can openers were clicked in place. Sealed in a bright red case and stamped with a silver cross and shield, another “Made in Switzerland” icon was born.

Carl Elsener Jr., the chief executive of Victorinox (pronounced vic-tor-EE-nox), which makes the famed pocket tool, smiled at the craft work and pointed proudly to a photo of the former American President George W. Bush and his wife, Barbara, touring the factory during a 1999 state visit and assembling their own knives by hand.

The United States has long been Victorinox’s most important market, Mr. Elsener said, raising his voice above the din of machinery. But after President Trump blindsided Switzerland last month by imposing a 39 percent import tariff, the highest for any Western country, the treasured ties were thrown into question.

“It was a shock,” said Mr. Elsener, who, like many Swiss executives, had anticipated an outcome more like that of the European Union, which negotiated a 15 percent tariff, or Britain, which was slapped with a basic tariff of 10 percent. “No one had expected such a drastic step,” he added.

After decades of easy entry into America, a wall has gone up for the Swiss, which for years faced a tariff of only 4.5 percent. If the higher tariffs stay in place, Victorinox will face a $13 million U.S. import tax bill next year. After keeping U.S. prices steady this year, Mr. Elsener may have to raise them. The company’s professional kitchen knives suddenly cost more than those of European competitors. And U.S. customs paperwork, once simple, is now a Kafkaesque ordeal.

“The new tariffs are hitting Switzerland’s export-oriented economy hard,” said Jan Atteslander, a director at EconomieSuisse, the lobbying group for Swiss businesses. “A swift agreement on reducing tariffs is essential.”