  We address dissipative dynamics of the one-dimensional nearest-neighbour $XX$
spin-$1/2$ chain governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)
equation. In the absence of dissipation the model is integrable. We identify a
broad class of dissipative terms that generically destroy integrability but
leave the operator space of the model fragmented into an extensive number of
dynamically disjoint subspaces of varying dimensions. In sufficiently small
subspaces the GKSL equation in the Heisenberg representation can be easily
solved, sometimes in a closed analytical form. We provide an example of such an
exact solution for a specific choice of dissipative terms. It is found that
observables experience the Wannier-Stark localization in the corresponding
operator subspace. As a result, the expectation values of the observables are
linear combinations of essentially a few discrete decay modes, the long time
dynamics being governed by the slowest mode. We examine the complex Liouvillian
eigenvalue corresponding to this latter mode as a function of the dissipation
strength. We find an exceptional point at a critical dissipation strength that
separates oscillating and non-oscillating decay. We also describe a different
type of dissipation that leads to a single decay mode in the whole operator
subspace. Finally, we point out that our exact solutions of the GKSL equation
entail exact solutions of the Schr\"odinger equation describing the quench
dynamics in closed spin ladders dual to the dissipative spin chains.
