Is it possible to have a Gas heavier than a liquid? Does such a pair of substances exist, that in certain physical conditions (temperature, pressure)  when both are placed in the same conditions, one will be a liquid, the other - a gas, and the gas density will be higher than that of the liquid?
(let us exclude special states of matter - e.g. don't count superfluid as liquid, or plasma as gas.)
 A: At its critical point of 318 K and 3.8 bar, SF$_6$ has a density of 742 kg/m$^3$.  This is certainly denser than some liquids. Of course we'd have to go for a slightly lower pressure to have an unambiguous gas.
Heptane is liquid at this temperature and pressure and has a density (at STP) of 697 kg/m$^3$.  This looks good but SF$_6$ is likely to be soluble in heptane given that it's "very soluble in ethanol, hexane, benzene" (wikipedia). Isopentane is lighter still but has problems with the boiling point and solubility. So we have to look for something more exotic.  
I can't find a phase diagram but pentaborane has a boiling point of 333 K and a density (again at STP) of 618 kg/m$^3$, meaning it will be liquid at the critical point of SF$_6$.  
I wouldn't fancy trying the experiment though: Pentaborane was found to be too dangerous to use as a rocket fuel, and it took decades to figure out a safe way to dispose of it.
A: I've just remembered that there was once a suggestion to use a mixture of xenon and oxygen under high pressure to allow people to float/fly/swim in it. It was also stated that water could be lighter than such a mixture.
According to Smithsonian Physical Tables the critical point for xenon is
$16.6\,\text{C}^{\circ},\quad 60\,\frac{\text{kg}}{\text{cm}^2},\quad 1.155\,\frac{\text{g}}{\text{cm}^3}$
while the density of water is around this conditions according to this online calculator is $1.0015 \frac{\text{g}}{\text{cm}^3}$ and it is still liquid ($60\,\frac{\text{kg}}{\text{cm}^2} \approx 58.84\,\text{bar}$).
One may have a look on phase diagrams of water and xenon on wolfram alpha:


*

*xenon phase diagram   

*water phase diagram
I suppose by lowering a bit the pressure under xenon critical point one can actually have a gas heavier than even water.
The original suggestion on high density xenon-oxygen breathable mixture should be in Ariadne section of New scientist, July 6, 1967. However this issue is not on Google books (yet?).
A: This is by now a very old question, but I think the video linked by @TejasKale really deserves its own answer, even if it is just for people not wanting to spend the time to watch it.
So it is a Youtube video by Cody'sLab titled Can You Float a Liquid on a Gas?, posted on the 15th of July 2017.
He tries to make several liquids float on high pressure xenon. He uses xenon because he can handle that with the equipment he has, while SF₆ would require much higher pressures.
The right conditions are created by cooling a test tube with the test liquid in liquid nitrogen and then adding gaseous xenon, which condenses. When enough liquid xenon has been collected he seals the tube and lets it heat up again so the xenon evaporates. So there isn't a good control over the exact pressure the xenon reaches.

*

*Plain water doesn't work because it forms a water-xenon clathrate or hydrate which is heavier than the xenon gas.


*A 75% ethanol-water mixture keeps the xenon out and does float on the xenon.


*Liquid sodium-potassium (NaK) also works. A little bit of added isopropryl alcohol helps to prevent an oxide coating forming on the NaK droplets and prevents them from sticking to the glass of the test tube.


*And with a bit of green colorant in the ethanol you have a basis to brew your own Frobscottle.
