  We prove that cluster observables of level-sets of the Gaussian free field on
the hypercubic lattice $\mathbb{Z}^d$, $d\geq3$, are analytic on the whole
off-critical regime $\mathbb{R}\setminus\{h_*\}$. This result concerns in
particular the percolation density function $\theta(h)$ and the (truncated)
susceptibility $\chi(h)$. As an important step towards the proof, we show the
exponential decay in probability for the capacity of a finite cluster for all
$h\neq h_*$, which we believe to be a result of independent interest. We also
discuss the case of general transient graphs.
