For the Bargmann--Fock field on $\mathbb R^d$ with $d\ge3$, we prove that the critical level $\ell_c(d)$ of the percolation model formed by the excursion sets $\{ f \ge \ell \}$ is strictly positive. This implies that for every $\ell$ sufficiently close to $0$ (in particular for the nodal hypersurfaces corresponding to the case $\ell=0$), $\{f=\ell\}$ contains an unbounded connected component that visits "most" of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behaviour of nodal hypersurfaces of these Gaussian fields in $\mathbb R^d$ for $d\ge3$ is very different from the behaviour of nodal lines of their two-dimensional analogues.