We present the first results from the Small Magellanic Cloud portion of a new Australia Telescope Compact Array HI absorption survey of both of the Magellanic Clouds, comprising over 800hr of observations. Our new HI absorption line data allow us to measure the temperature and fraction of cold neutral gas in a low-metallicity environment. We observed 22 separate fields, targeting a total of 55 continuum sources, against 37 of which we detected HI absorption; from this we measure a column-density-weighted mean average spin temperature of <Ts>=150K. Splitting the spectra into individual absorption line features, we estimate the temperatures of different gas components and find an average cold gas temperature of ~30K for this sample, lower than the average of ~40K in the Milky Way. The HI appears to be evenly distributed throughout the SMC, and we detect absorption in 67% of the lines of sight in our sample, including some outside the main body of the galaxy (N_HI_>2x10^21^cm^-2^). The optical depth and temperature of the cold neutral atomic gas show no strong trend with location spatially or in velocity. Despite the low-metallicity environment, we find an average cold gas fraction of ~20%, not dissimilar from that of the Milky Way.