Yes — you can lose Trophies, especially in the higher Arenas. If you finish low in a daily round, you’ll drop some Trophies, which can slow your progress. The higher you climb, the tougher the competition — and the bigger the risk of losing ground.
However, once you’ve unlocked a new Arena on the Trophy Road, you can’t be demoted. Your Trophy Road progression is permanent, ensuring you always keep access to the Arenas you’ve reached.