We consider the following problem:
\begin{align*}
-\Delta_{\mathbb{B}^N} u \, - \, \lambda u &= |u|^{p-1}u, \quad u \in H^{1}\(\mathbb{B}^N\),
\end{align*}
where $\mathbb{B}^N$ represents the Poincar\`e ball model of the hyperbolic space, $1 < p < 2^* - 1 = \frac{N+2}{N-2}$, $\lambda < \frac{(N-1)^2}{4}$, $N \geq 4$. Here, we extend the results by Mancini, Sandeep [2008] and Bhakta, Sandeep [2012], which establish the existence of radial positive and radial sign-changing solutions, respectively. We prove the existence and multiplicity of non-radial sign-changing solutions. To prove the existence of such solutions, we consider two isometric group actions on $\mathbb{B}^N$ and pose a variational problem on a suitable subspace of $H^{1}\(\mathbb{B}^N\)$. We solve the variational problem in two cases, depending on the fixed point set of the groups.