  We study an ergodic problem associated to a non-local Hamilton-Jacobi
equation defined on the whole space $\lambda-\mathcal{L}[u](x)+|Du(x)|^m=f(x)$
and determine whether (unbounded) solutions exist or not. We prove that there
is a threshold growth of the function $f$, that separates existence and
non-existence of solutions, a {phenomenon} that does not appear in the local
version of the problem. Moreover, we show that there exists a critical ergodic
constant, $\lambda_*$, such that the ergodic problem has solutions for
$\lambda\leq \lambda_*$ and such that the only solution bounded from below,
which is unique up to an additive constant, is the one associated to
$\lambda_*$.
