Clumsy Balloon Slime

Only one exists in a world at a time and it drifts around the space layer in any weather, so the search is a single sweep along the top of the map rather than a hunt. Before hardmode it refuses to spawn in the middle strip of the world, which is why a sky bridge built straight out from spawn can go hundreds of hours without turning one up. Push the bridge toward an ocean instead, or wait for hardmode, which lifts that restriction entirely. A Lifeform Analyzer removes the rest of the guesswork.

It cannot hurt you and it cannot be killed. The balloon is the part you attack: pop it and the passenger drops away as a Clumsy Slime you can move into a house. There is no reason to wait, though the floating version stops appearing in that world afterward. If the town version later dies it comes back on its own, so you never need to go looking a second time.