  We study the effect of hole doping on the Kitaev spin liquid (KSL) and find
that for ferromagnetic (FM) Kitaev exchange $K$ the system is very susceptible
to the formation of a FM spin polarization. Through density matrix
renormalization group (DMRG) simulations on finite systems, we uncover that the
introduction of a single hole with a hopping strength of just $t\sim{}0.28K$ is
enough to disrupt fractionalization and polarize the spins in the [001]
direction due to an order-by-disorder mechanism. Taking into account a material
relevant FM anisotropic spin exchange $\varGamma$ drives the polarization
towards the [111] direction via a reorientation transition into a topological
FM state with chiral magnon excitations. We develop a parton mean-field theory
incorporating fermionic holons and bosonic spinons/magnons, which accounts for
the doping induced FM phases and topological magnon excitations. We discuss
experimental signatures and implications for Kitaev candidate materials.
