For a thin ferromagnetic film with the Dzyaloshinskii-Moriya interaction (DMI), we derive an expression of the extrinsic ferromagnetic resonance (FMR) linewidth in a quantum-mechanical way, taking into account scatterings from structural inhomogeneity. In the presence of the DMI, the magnon dispersion exhibits rich resonant states, especially in small external magnetic fields and strong DMI strength. It is found that the FMR linewidth shows several characteristic features such as a finite linewidth at zero frequency and peaks in the low-frequency range.