A solid can be lighter than air and still not float

Silica aerogel can be made with a solid network less dense than air. It does not float away, because its pores are full of air.

Samuel Kistler made the first aerogels in 1931 by replacing the liquid in a gel with gas without collapsing the gel's skeleton. Modern silica aerogels can have densities around a few milligrams per cubic centimetre, below the 1.2 milligrams of a cubic centimetre of air. The record-holders are even lighter.

The skeleton alone would be buoyant in air. The object you hold is skeleton plus the air in the pores, and that sum is heavier than the air it displaces, so it sits in the hand. Evacuate the pores and the skeleton collapses, or, if it could be held open in a vacuum, it would be a different and much more fragile experiment. Stardust, the NASA probe, used aerogel to catch comet dust. The dust buried itself in a solid that is mostly nothing.