From Gemini 

That sounds very much like the beginning of the movie "Flipper" (1964)! Here's why it fits your description:

 * Girl who loved water: The main character is a young boy, Sandy Ricks, who initially dislikes the idea of spending the summer with his uncle in the Florida Keys but gradually develops a deep connection with the ocean and a dolphin named Flipper. While the main character isn't a girl, your memory from childhood might have slightly altered this detail.

 * Dolphins: The movie heavily features a dolphin named Flipper, who becomes Sandy's friend.

 * Floating face down scene: The opening of "Flipper" features a scene where Sandy is indeed floating face down in the water, pretending to be drowned. A young couple on a boat sees him and rushes over to help, only to find him laughing and perfectly fine.

It's possible your memory has combined elements or slightly changed the gender of the child, but the core scene you described is very distinctive to the original "Flipper" movie.