A light-weight, flexible electromagnetic interference (EMI) shield was prepared by creating a layer-structured metal-polymer composite film consisting of electrospun nylon 66 nanofibers with silver films. The EMI shielding effectiveness (SE), specific SE, and absolute SE of the composite were as high as 60.6 dB, 67.9 dB cm3/g, and 6792 dB cm2/g in the X- and Ku-bands, respectively. Numerical and analytical calculations suggest that the energy of EM waves is predominantly absorbed by inter-layer multiple reflections. Because the absorbed EM energy is dissipated as heat, the thermal conductivity of absorption-dominant EMI shields is highly significant. Measured thermal conductivity of the composite was found to be 4.17 Wm−1K−1 at room temperature, which is higher than that of bulk nylon 66 by a factor of 16.7. The morphology and crystallinity of the composite were examined using scanning electron microscopy and differential scanning calorimetry, respectively. The enhancement of thermal conductivity was attributed to an increase in crystallinity of the nanofibers, which occurred during the electrospinning and subsequent hot pressing, and to the high thermal conductivity of the deposited silver films. The contribution of each fabrication process to the increase in thermal conductivity was investigated by measuring the thermal conductivity values after each fabrication process.