The purpose of this paper is to establish the Donsker–Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier–Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in [17] in a discrete-time setting and extended in [26] to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman–Kac semigroup, which we verify by using Malliavin calculus.