  We study the effects of doping a Mott insulator on the honeycomb lattice
where spins interact via direction dependent Kitaev couplings J_K, and weak
antiferromagnetic Heisenberg couplings J. This model is known to have a spin
liquid ground state and may potentially be realized in correlated insulators
with strong spin orbit coupling. The effect of hole doping is studied within a
t-J-J_K model, treated using the SU(2) slave boson formulation, which correctly
captures the parent spin liquid. We find superconductor ground states with spin
triplet pairing that spontaneously break time reversal symmetry. Interestingly,
the pairing is qualitatively different at low and high dopings, and undergoes a
first order transition with doping. At high dopings, it is smoothly connected
to a paired state of electrons propagating with the underlying free particle
dispersion. However, at low dopings the dispersion is strongly influenced by
the magnetic exchange, and is entirely different from the free particle band
structure. Here the superconductivity is fully gapped and topological,
analogous to spin polarized electrons with px+ipy pairing. These results may be
relevant to honeycomb lattice iridates such as A2IrO3 (A=Li or Na) on doping.
