This is a lovely resort area of the island, with a sunset and hotel. Well, there was a hotel before it turned into goo. Do NOT TALK TO ANYONE. First clear out all the goo everywhere. It'll make the next part easier.

Stand in the center of the concrete area and talk to the hotel owner. As soon as you do, a flat manta ray shadow will come at you. Shoot it to break it into 2 smaller ones. Soon you will end up with TRILLIONS of little manta ray shadows everywhere. Shoot them from up on trees usually, that's safest. They leave little slime trails behind you. The bars have coins if you get hurt.

Soon they'll all turn dark pink and hostile. Now head to the beach and just shoot them as they come at you. To the left of the beach are some boxes with coins plus the center one has a one-up mushroom. Don't worry about the slime, once you get the last one, it goes away. And the hotel comes up, and the sprite appears!

Tip from joe reedy:
Before they all come after you hide under a hut to spray them.

Super Mario Sunshine Walkthrough