JWST/NIRCam obtained high angular resolution (0.05-0.1 arcsec), deep near-infrared 1-5 μm imaging of Supernova (SN) 1987A taken 35 yr after the explosion. In the NIRCam images, we identify: (1) faint H2 crescents, which are emissions located between the ejecta and the equatorial ring, (2) a bar, which is a substructure of the ejecta, and (3) the bright 3-5 μm continuum emission exterior to the equatorial ring. The emission of the remnant in the NIRCam 1-2.3 μm images is mostly due to line emission, which is mostly emitted in the ejecta and in the hotspots within the equatorial ring. In contrast, the NIRCam 3-5 μm images are dominated by continuum emission. In the ejecta, the continuum is due to dust, obscuring the centre of the ejecta. In contrast, in the ring and exterior to the ring, synchrotron emission contributes a substantial fraction to the continuum. Dust emission contributes to the continuum at outer spots and diffuse emission exterior to the ring, but little within the ring. This shows that dust cooling and destruction time-scales are shorter than the synchrotron cooling time-scale, and the time-scale of hydrogen recombination in the ring is even longer than the synchrotron cooling time-scale. With the advent of high sensitivity and high angular resolution images provided by JWST/NIRCam, our observations of SN 1987A demonstrate that NIRCam opens up a window to study particle-acceleration and shock physics in unprecedented details, probed by near-infrared synchrotron emission, building a precise picture of how an SN evolves.