  Exactly solvable Hamiltonians with spin liquid ground states have proven to
be extremely useful, not only because they unambiguously demonstrate that these
phases can arise in systems of interacting spins but also as a pedagogical
illustration of the concept and as a controlled starting point for further
theoretical analysis. However, adding dissipative couplings to the environment
- an important aspect for the realization of these phases - generically spoils
the exact solvability. We here present and study a Lindbladian, describing a
square-lattice spin-liquid with dissipative coupling to the environment, that
admits an exact solution in terms of Majorana fermions coupled to static
$\mathbb{Z}_2$ gauge fields. This solution allows us to characterize the
steady-state solutions as well as ``quasiparticle'' excitations within the
Lindbladian spectrum. This emergence of distinct types of quasiparticle
excitations of the Lindbladian leads to a separation of timescales that govern
the equilibration time of the expectation values of different classes of
observables, some of which we identify as fractionalized string-like operators.
This exactly solvable Lindbladian is expected to provide a starting point for a
better understanding of the behavior of fractionalized systems under
dissipative time evolution.
