For the Bargmann--Fock field on Rd with d>2, we prove that the critical level lc(d) of the percolation model formed by the excursion sets {f≥l} is strictly positive. This implies that for every l sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case l=0), {f=l} contains an unbounded connected component that visits "most" of the ambient space. Our findings (joint work with A. Rivera, PF Rodriguez, and H. Vanneuville) actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. A consequence of this result combined with a result of A. Rivera is that Monochromatic Random Wave in high dimensions percolates almost surely.