We present new imaging observations of the remnant of Supernova (SN) 1987A at 44 GHz, performed in 2011 with the Australia Telescope Compact Array (ATCA). The 035×023 resolution of the diffraction-limited image is the highest achieved to date in high-dynamic range. We also present a new ATCA image at 18 GHz derived from 2011 observations, which is super-resolved to 025. The flux density is 40±2 mJy at 44 GHz and 81±6 mJy at 18 GHz. At both frequencies, the remnant exhibits a ring-like emission with two prominent lobes, and an east-west brightness asymmetry that peaks on the eastern lobe. A central feature of fainter emission appears at 44 GHz. A comparison with previous ATCA observations at 18 and 36 GHz highlights higher expansion velocities of the remnant eastern side. The 18-44 GHz spectral index is α=−0.80 (Sν∝να
Nearby in the stack
). The spectral index map suggests slightly steeper values at the brightest sites on the eastern lobe, whereas flatter values are associated with the inner regions. The remnant morphology at 44 GHz generally matches the structure seen with contemporaneous X-ray and H
α
observations. Unlike the H
α
emission, both the radio and X-ray emission peaks on the eastern lobe. The regions of flatter spectral index align and partially overlap with the optically-visible ejecta. Simple free-free absorption models suggest that emission from a pulsar wind nebula or a compact source inside the remnant may now be detectable at high frequencies, or at low frequencies if there are holes in the ionised component of the ejecta.