U.S. farmers, already losing sales to China, are facing new threats to sales in other big overseas markets as trade tensions spread globally.

Mexico this week imposed tariffs on major U.S. exports such as cheese and pork, while Canada and the European Union are considering tariffs on imports of U.S. food and farm goods from corn to orange juice to peanut butter, in response to the U.S. placing tariffs last week on steel and aluminum imports from those countries. In addition, analysts say, China could target other crops and...