A traditional quantum mechanical system of $N$ particles has a finite number of $3N$ position coordinates as degrees of freedom. The Hamiltonian can be expressed for any $N$; hence one can form the corresponding system where $N$ is treated as uncertain. This is done by working in the corresponding Fock space, defined as the direct sum of all $N$-particle Hilbert spaces. The resulting system has now an infinite number of degrees of freedom. That this constitutes a nonrelativistic QFT becomes apparent when changing the notation to that of second quantization, with creation and annihilation operators that change the number of particles.
Thus all statistical mechanics done in the grand canonical ensemble is nonrelativistic quantum field theory.
Other examples are given by the statistical mechanics of crystals, which are infinitely extended periodic systems. The periodicity, however, allows one to map the problem on the 3-dimensional torus, which is a compact manifold; so that the total number of particles on the torus is finite (but huge).
Both classes of examples together constitute the main examples of nonrelativistic quantum field theories.
In general, one considers the infinite number of degrees of freedom as the essence of a quantum field theory, but this is the case only when the underlying position space is non-compact. In the compact case, and in particular for a torus (e.g. crystal) or a single point (= 0-dimensional field theory; e..g, ordinary quantum mechanics in the rest frame in a basis of harmonics), there are only finitely many degrees of freedom.
0-dimensional QFT is QM in the same sense as 0-dimensional classical field theory is described by ODEs rather than PDEs for higher dimensions.
The difference between relativistic and nonrelativistic is the same in QFT as in classical field theory - namely whether the theory is invariant under the Poincare group or the Galilei group.