For closed convex subsets D of a Banach spaces, in 2009, Tomonari Suzuki [11] proved that the fixed point property (FPP) for nonexpansive mappings and the FPP for nonexpansive semigroups are equivalent. In this paper some relations between the aforementioned properties for mappings and semigroups defined on D, a closed convex subset of the hyperbolic metric space (D,ρ), are studied. This work arises as a generalization to the space (D,ρ) of the study made by Suzuki. Keywords: ρ-nonexpansive mappings, fixed point property, semigroups. To cite this article: L. Benitez-Babilonia, Fixed point property for nonexpansive mappings and nonexpansive semigroups on the unit disk, Rev. Integr. Temas Mat. 33 (2015), no. 1, 41-50.