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We study an ergodic problem associated to a non-local Hamilton-Jacobiequation 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 thereis a threshold growth of the function $f$, that separates existence andnon-existence of solutions, a {phenomenon} that does not appear in the localversion of the problem. Moreover, we show that there exists a critical ergodicconstant, $\lambda_*$, 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$\lambda_*$.

On unbounded solutions of ergodic problems for non-local Hamilton-Jacobi equations