Sulfur Hexafluoride is 6 times denser than Earth's air, but far less dense than water. It is possible that on a high gravity planet with a cold and quiet atmosphere it would settle out and form the gaseous sea you are talking about. But it would have to be very cold and calm.
There are a couple of problems. You may be familiar with density towers of liquids. Liquid density towers separate based on density, but their separation is held together by cohesive forces in the liquid. You need liquids that won't quickly diffuse and mix into one another. Oil and water do the trick. The density tower I linked uses various forms of sugar water (honey and corn syrup) because they are very viscous and diffuse slowly. It also tells you to be very careful to not agitate or mix your liquids up. They will not separate very fast.
For gasses, density separation is theoretically possible, but more difficult. Gasses diffuse easily and there are no cohesive forces holding them together. They have a low viscosity and mix more easily than liquids. On Earth, sulfur hexafluoride does not separate out of our atmosphere despite the fact that it is far more dense.
As temperatures become colder, viscosity increases, and diffusion decreases. Therefore, density separation with gasses is easier at lower temperatures. This becomes more plausible on an ice planet.
Another factor is gravity. In order for the gasses to separate, the difference in density has to win the battle against diffusion. When gravity is higher the difference in density will matter more. On Earth we occasionally use centrifuges to separate isotopes in gaseous form. The spinning centrifuges simulate a very high gravity. The density differences of isotopes are slight, so a long series of centrifuges has to run at very high speeds to overcome that slight difference.
The other factor is weather. If your planet does have separated gas, a good storm can mix it back up again. Wind is not your friend.
To summarize, this planet becomes more plausible if you have a cold planet with high gravity, and no violent weather. On an Earth-like planet, even the heaviest gasses will not separate. I have no idea how cold, high gravity, or quiet your planet would have to be.