We derive the exact S-matrix for the scattering of particular representations of the centrally-extended p s u 1 | 1 2 $$ \mathfrak{p}\mathfrak{s}\mathfrak{u}{\left(1\Big|1\right)}^2 $$ Lie superalgebra, conjectured to be related to the massive modes of the light-cone gauge string theory on AdS 2 × S 2 × T 6 . The S-matrix consists of two copies of a centrally-extended p s u 1 | 1 $$ \mathfrak{p}\mathfrak{s}\mathfrak{u}\left(1\Big|1\right) $$ invariant S-matrix and is in agreement with the tree-level result following from perturbation theory. Although the overall factor is left unfixed, the constraints following from crossing symmetry and unitarity are given. The scattering involves long representations of the symmetry algebra, and the relevant representation theory is studied in detail. We also discuss Yangian symmetry and find it has a standard form for a particular limit of the aforementioned representations. This has a natural interpretation as the massless limit, and we investigate the corresponding limits of the massive S-matrix. Under the assumption that the massless modes of the light-cone gauge string theory transform in these limiting representations, the resulting S-matrices would provide the building blocks for the full S-matrix. Finally, some brief comments are given on the Bethe ansatz.