The author studies rigidity properties for lattices in the group \(G_p=\mathrm{PU}(p,1)\) or \(\mathrm{PS}p(p,1)\). The case of \(\mathrm{PO}(p,1)\) has been considered by M. Bucher et al. [Springer INdAM Ser. 3, 47–76 (2013; Zbl 1268.53056)] and also by S. Francaviglia and B. Klaff [Geom. Dedicata 117, 111–124 (2006; Zbl 1096.51004)]. This study originates in the results of Mostow about rigidity properties of lattices in locally symmetric spaces. Let \(X^p\) be the hyperbolic space associated to \(G_p\) and \(\Gamma \subset G_p\) a lattice such that the manifold \(M^p=\Gamma \backslash X^p\) is non-compact and has finite volume. For a representation \(\rho \) of \(\Gamma \) in \(G_m\), \(m\geq p\), the volume \(\mathrm{Vol}(\rho )\) is defined as the infimum of volumes of submanifolds in \(X^m\) associated to certain \(\rho \)-equivariant maps. Assume that \(\Gamma \) is without torsion. It is proven that \(\mathrm{Vol}(\rho )\leq\mathrm{Vol}(M^p)\) and equality holds if the representation \(\rho \) is a faithful representation of \(\Gamma \) into the isometry group of a totally geodesic copy of \(X^p\) contained in \(X^m\). Furthermore, the author considers a sequence \(\rho _n\) of representations of \(\Gamma \) in \(G_m\) such that \[\lim _{n\to \infty }\mathrm{Vol}(\rho _n)=\mathrm{Vol}(M^p).\] Then it is shown that there are \(g_n\in G_m\) such that \(g_n\circ \rho _n\circ g_n^{-1}\) converges to a representation which preserves a totally geodesic copy of \(X^p\) and whose \(X^p\) component is conjugated to the standard lattice embedding of \(\Gamma \) into \(G_p\).