Although nylon is a bit denser than water, it floats and even larger pieces that are larger than micro-fibbets are commonly used in dry fly fishing. Hint - what are monofilament leaders made of?
+1 on Calftail's answer of surface tension
You can even float a needle on water because of surface tension. Look at the video below which shows the effect of surface tension. I chose it because it also shows the effect of surfactants
= SURFACe ACTive ageNTS. The soap is a surfactant which destroys the surface tension and allows the needle to sink.
The reason that surface tension exists is that water H2O is a polar molecule with a positive pole of the hydrogen atoms and the negative pole of oxygen. Water molecules are miniature bar magnets that link up and it is these weak molecular links that form surface tension.
Surface tension is also why water beads up on a surface.
As an aside it is this same polar quality of water that creates MRI images based on flipping these molecules in a strong magnetic field. That is why MRI machines use superconducting magnets. It is to force the alignment of all the water molecules in our bodies.
We use the polar quality of water to our advantage when we put on floatant. A floatant is a non polar molecule that repels water.