  We consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$, for
$d\geq3$, and give sharp bounds on the probability that the radius of a finite
cluster in the excursion set $\{\varphi \geq h\}$ exceeds a large value $N$,
for any height $h \neq h_*$, where $h_*$ refers to the corresponding
percolation critical parameter. In dimension $d=3$, we prove that this
probability is sub-exponential in $N$ and decays as
$\exp\{-\frac{\pi}{6}(h-h_*)^2 \frac{N}{\log N} \}$ as $N \to \infty$ to
principal exponential order. When $d\geq 4$, we prove that these tails decay
exponentially in $N$. Our results extend to other quantities of interest, such
as truncated two-point functions and the two-arms probability for annuli
crossings at scale N.
