In this article, we study the excursion sets 𝒟 p = f - 1 ( [ - p , + ∞ [ ) where f is a natural real-analytic planar Gaussian field called the Bargmann–Fock field. More precisely, f is the centered Gaussian field on ℝ 2 with covariance ( x , y ) ↦ exp ( - 1 2 | x - y | 2 ) . Alexander has proved that, if p ≤ 0 , then a.s. 𝒟 p has no unbounded component. We show that conversely, if p > 0 , then a.s. 𝒟 p has a unique unbounded component. As a result, the critical level of this percolation model is 0 . We also prove exponential decay of crossing probabilities under the critical level. To show these results, we rely on a recent box-crossing estimate by Beffara and Gayet. We also develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.