We present two new radio continuum images from the Australian Square Kilometre Array Pathfinder (ASKAP) survey in the direction of the Small Magellanic Cloud (SMC). These images are part of the Evolutionary Map of the Universe (EMU) Early Science Project (ESP) survey of the Small and Large Magellanic Clouds. The two new source lists produced from these images contain radio continuum sources observed at 960 MHz (4489 sources) and 1320 MHz (5954 sources) with a bandwidth of 192 MHz and beam sizes of 30.0x30.0 and 16.3x15.1, respectively. The median Root Mean Squared (RMS) noise values are 186$mu$Jy beam$^{-1}$ (960 MHz) and 165$mu$Jy beam$^{-1}$ (1320 MHz). To create point source catalogues, we use these two source lists, together with the previously published Molonglo Observatory Synthesis Telescope (MOST) and the Australia Telescope Compact Array (ATCA) point source catalogues to estimate spectral indices for the whole population of radio point sources found in the survey region. Combining our ASKAP catalogues with these radio continuum surveys, we found 7736 point-like sources in common over an area of 30 deg$^2$. In addition, we report the detection of two new, low surface brightness supernova remnant candidates in the SMC. The high sensitivity of the new ASKAP ESP survey also enabled us to detect the bright end of the SMC planetary nebula sample, with 22 out of 102 optically known planetary nebulae showing point-like radio continuum emission. Lastly, we present several morphologically interesting background radio galaxies.
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