Eliezer: the Hilbert space of QM is generally believed to have a countable basis

It would be more accurate to say that when a Hilbert space is used, it has countable dimension. But the Hilbert space of a quantum field, naively, ought to have uncountable dimension, because there are continuum-many degrees of freedom. In practice, quantum field theory is done using a lot of formalism which only formally refers to an underlying Hilbert space - sum over histories, operator algebras - and even the simplest real-life quantum field theories (i.e. those used in particle physics) have not been realized with mathematical rigor, in terms of a particular Hilbert space.

When you get to quantum gravity, things change again, because one now has independent arguments for there being only a finite number of degrees of freedom in any finite volume of space. But the attempts to formulate the theory in such terms from the beginning are still rather preliminary.