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\rangle \text- |1\rangle channel at 159 G (1 G = 10^–4 T). A controllable population of atoms in the spin state |2\rangle is introduced into a spin-polarized |1\rangle gas to investigate the coupling between a nonresonant spin component and the resonant inelastic dynamics. In the magnetic-field range studied here, the |2\rangle atoms are far detuned from the |1\rangle \text- |2\rangle and |2\rangle \text- |2\rangle p-wave resonances, and therefore do not directly participate in the resonant two-body scattering between two |1\rangle atoms. Nevertheless, component-resolved loss spectra reveal that the |2\rangle component exhibits a double-peak loss structure at the same magnetic-field positions as the dipole-split p-wave resonance peaks of the |1\rangle component. This spectral correspondence is consistent with a loss process mediated by intermediate quasibound molecular states associated with the resonant |1\rangle \text- |1\rangle channel. Time-resolved measurements further show that the decay of the |2\rangle atoms is described by an effective |1\rangle \text- |1\rangle \text- |2\rangle three-body loss term proportional to n_1^2 n_2 , whereas models based on |1\rangle \text- |2\rangle two-body loss or |1\rangle \text- |2\rangle \text- |2\rangle three-body loss do not reproduce the measured dynamics. The observations are described within a cascade atom-dimer model, in which two |1\rangle atoms first form a quasibound p-wave dimer d_11 , followed by inelastic relaxation through either the a_1 \text- d_11 or the a_2 \text- d_11 channel. By varying the initial population ratio N_2/N_1 , we find that the extracted loss coefficients L_111 and L_112 show opposite overall trends, while the weighted coefficient L_111+2N_2L_112/(3N_1) 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 tool for distinguishing atom-dimer-mediated cascade loss from direct three-body recombination.