Does this apply for solid bodies, too?

Yes, every kind of body is affected by buoyancy:
If you fill the balloon with water or some kind of other material, the following will happen:

The balloon with a small volume is affected by little buoyancy. Due to the fact that its weight force is much more greater it will sink to the ground. The big balloon displaces more of the air and thus its buoyancy is greater.
For the reason that the bigger balloon is made out of more material, it will also sink to the ground, because therefore its weight force is greater, too. But the buoyancy is so small that one needs very sensitive measuring instruments to prove this effect.