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In this paper, we study nonlinear Helmholtz equations (NLH) $-\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u$ in $\mathbb{H}^N$, $N\geq 2$ where $\Delta_{\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in the hyperbolic space $\mathbb{H}^N$ and $\Gamma\in L^\infty(\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $\lambda>0$ and $p>2$. The oscillatory behaviour and decay rates of radial solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle for the Helmholtz operator in $\mathbb{H}^N$. As a byproduct, we obtain simple counterexamples to certain Strichartz estimates.