Thin Ice (1937)

Lili is a skating instructor at a grand hotel in the Alps. An international conference is booked at the hotel. The conference is led by Prince Rudolph, whose plan is really to keep a pair of feuding countries at odds with each other. Feining illness, the Prince moves into a small inn so he can enjoy some skiing in private -- and delay the conference. One morning he meets Lili on the slopes and they hit it off; but she has no idea her "Rudy" is the Prince. That evening Lili is seen leaving the Prince's car, having been given a ride home by her beau, a cousin of the Prince's chauffeur. Tongues wag and Lili is thought to be romantically involved with the Prince. This gets her lots of attention and a starring role in her own ice skating revue. But when she finds out people think she is involved with the Prince she is horrified, while Rudy is amused and plays along.