  For a large family of stationary continuous Gaussian fields $f$ on
$\mathbb{R}^d$, including the Bargmann-Fock and Cauchy fields, we prove that
there exists at most one unbounded connected component in the level set
$\{f=\ell\}$ (as well as in the excursion set $\{f\geq\ell\}$) almost surely
for every level $\ell\in \mathbb{R}$, thus proving a conjecture proposed by
Duminil-Copin, Rivera, Rodriguez & Vanneuville. As the fields considered are
typically very rigid (e.g.~analytic almost surely), there is no sort of finite
energy property available and the classical approaches to prove uniqueness
become difficult to implement. We bypass this difficulty using a soft shift
argument based on the Cameron-Martin theorem.
