Yes – when magma cools, it forms a type of rock called igneous rock. The inside of this rock is affected by how much gas escapes during the eruption, which means that some rocks like pumice (as Carl mentioned) can be so light that it floats!
Generally, no – rock is much more dense than water. The only real exception (as mentioned) is Pumice, which has a very high content of void space filled with gas. But pumice is only a small proportion of rock formed – it forms when you cool lava really, really quickly, by quenching it in water, for example.
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