I will note that even in QM, both space and time are considered continuous - the Planck length is just a smallest measurable distance, but nothing in QM currently assumes that particles must be separated by an integer multiple of the Planck length (unlike spins, for example).

I believe there are some theories of quantum gravity that do rely on the idea that space-time is quantized in integer multiples of Planck's length, but these are far from definitive theories.

A much more relevant limit in terms of possible information density is Heisenberg's uncertainty principle, which essentially puts a limit on the maximum possible precision for any measurement.