PLYMOUTH, Wis. — It’s a common observation here that you can’t turn off the cows — not for Christmas, and not for a trade war.

So as President Trump’s aggressive trade measures prompt other countries to retaliate with barriers to American goods, dairy farmers and cheesemakers in the rolling, bright green hills of Wisconsin are growing anxious about what will happen to all of the milk and cheese they churn out and typically sell overseas.

“If export markets get shut off, I could see us getting to the point where we’re dumping our milk in the fields,” said Jeff Schwager, the president of Sartori Company, which has produced cheese in a nearby town for generations with milk it purchases from more than 100 dairy farms throughout Wisconsin. “It’ll be a big ripple effect through the state.”

Mr. Trump has set off trade clashes with countries around the world, demanding new trade agreements and slapping tariffs on allies to reset what he says are deeply unfair terms that hurt American companies and workers. He has singled out certain American industries that he says are at a global disadvantage, including automobiles and dairy — which led to a public spat between Mr. Trump and the Canadian prime minister Justin Trudeau over Canada’s dairy tariffs.