Motivated by the growing interest in spin liquids and topological phases, as well as the rise of deformation engineering, we study the combined effects of deformation and magnetic fields on the honeycomb Kitaev model. The Kitaev model, as one of the prototypical and exactly solvable spin liquid-hosting models, serves as a simple platform that demonstrates the rich physics one can expect at the intersection of deformation physics and quantum spin liquids. Our work builds on a simplified solution to the undeformed base model that we present. This simplified solution allows for a straightforward extension of our analysis to the deformed case. After incorporating periodic deformations into the Kitaev model (chosen for its similarity to moir'e physics), we investigate the effects of a hexagonally symmetric deformation on the band structure. We find that deformation leads to a smaller Brillouin zone with new band gaps at the edges, indicating the potential for topological transitions. Finally, we introduce a magnetic field to break time-reversal symmetry and thereby allow for non-trivial topology. We find that, under specific parameter conditions, the magnetic field leads to multiple band-gap closings and openings. An investigation into topological properties reveals nontrivial Chern numbers and a plethora of topological transitions. The resulting nontrivial band topology may motivate future investigations of thermal Hall or related transverse responses, which are not calculated here. We also propose a bulk-measurement approach to the Chern numbers and a possible path to their physical realization. Most importantly, our results demonstrate the rich phenomenology that can arise from the interplay between deformation and spin-liquid physics.