So, on a previous question of mine, someone answered [that it would be feasible to build a city on a waterborne floating island using a pumice layer below the ground](https://worldbuilding.stackexchange.com/questions/64252/ferrying-an-island-with-mortals-on-top-is-it-possible-without-killing-them).

That got me thinking about the possibility of having a city built on a floating island, but airborne. 

Now, I know that there have been questions about floating islands already... but those implicated major geological catastrophes projecting rocks into the air, technological or magical human intervention and atmospheric gases different than our own.

But my question is different: Is it possible to conceive of an imaginary rock that would float on the air of an Earth-like atmosphere, some distance above the ground? I know this seems impossible, but having a rock that floats on water also seems impossible and we have pumice.

I don't want real world examples (they don't exist), only to know if such a rock could be theoretically possible. A geological material that could form floating islands in the sky where the building of cities would be possible.