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We consider the Hardy–Schrödinger operator Lγ:=-Δ𝔹n-γ⁢V2{L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space 𝔹n{\mathbb{B}^{n}} (n≥3{n\geq 3}). Here V2{V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V2⁢(r)∼1r2{V_{2}(r)\sim\frac{1}{r^{2}}}. As in the Euclidean setting, Lγ{L_{\gamma}} is positive definite whenever γn-2n-4⁢(n⁢(n-4)4-γ){\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}. On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.