We report an experimental study of cross-component three-body loss in an ultracold fermionic gas of ⁶Li near the narrow p-wave Feshbach resonance in the |1〉-|1〉 channel at 159 G. A controllable population of atoms in the spin state |2〉 is introduced into a spin-polarized |1〉 gas to investigate the coupling between a nonresonant spin component and the resonant inelastic dynamics. In the magnetic-field range studied here, the |2〉 atoms are far detuned from the |1〉-|2〉 and |2〉-|2〉 p-wave resonances, and therefore do not directly participate in the resonant two-body scattering between two |1〉 atoms. Nevertheless, componentresolved loss spectra reveal that the |2〉 component exhibits a double-peak loss structure at the same magnetic-field positions as the dipole-split p-wave resonance peaks of the |1〉 component. This spectral correspondence is consistent with a loss process mediated by intermediate quasibound molecular states associated with the resonant |1〉-|1〉 channel. Time-resolved measurements further show that the decay of the |2〉 atoms is described by an effective |1〉-|1〉-|2〉 three-body loss term proportional to n₁²n₂, whereas models based on |1〉-|2〉 two-body loss or |1〉-|2〉-|2〉 three-body loss do not reproduce the measured dynamics. The observations are described within a cascade atom-dimer model, in which two |1〉 atoms first form a quasibound p-wave dimer d₁₁, followed by inelastic relaxation through either the a₁-d₁₁ or the a₂-d₁₁ channel. By varying the initial population ratio N₂/N₁, we find that the extracted loss coefficients L₁₁₁ and L₁₁₂ show opposite overall trends, while the weighted coefficient L₁₁₁+2N₂L₁₁₂/(3N₁) remains approximately unchanged within the experimental uncertainty. This behavior indicates that the nonresonant spin component mainly redistributes atom-dimer relaxation between competing decay channels, rather than substantially modifying the initial formation step of the quasibound dimer. These results provide component-resolved evidence for a cascade loss mechanism near a narrow p-wave Feshbach resonance and suggest a diagnostic for distinguishing atom-dimer-mediated cascade loss from direct threebody recombination.