Star-forming galaxies are thought to dominate the sub-mJy radio population, but recent work has shown that low-luminosity active galactic nuclei (AGNs) can still make a significant contribution to the faint radio source population. Spectral indices are an important tool for understanding the emission mechanism of the faint radio sources. We have observed the extended Chandra Deep Field South at 5.5 GHz using a mosaic of 42 pointings with the Australia Telescope Compact Array. Our image reaches an almost uniform sensitivity of ~12 μJy rms over 0.25 deg2 with a restoring beam of 4.9 arcsec × 2.0 arcsec, making the ATLAS 5.5 GHz survey one of the deepest 6 cm surveys to date. We present the 5.5 GHz catalogue and source counts from this field. We take advantage of the large amount of ancillary data in this field to study the 1.4 to 5.5 GHz spectral indices of the sub-mJy population. For the full 5.5 GHz selected sample we find a flat median spectral index, αmed = -0.40, which is consistent with previous results. However, the spectral index appears to steepen at the faintest flux density levels (S5.5 GHz < 0.1 mJy), where αmed= -0.68. We performed stacking analysis of the faint 1.4 GHz selected sample (40<S1.4 GHz<200 μJy) and found a steep average spectral index, α = -0.8, consistent with synchrotron emission. We also found a weak trend of steepening spectral index with redshift. Several young AGN candidates are identified using spectral indices, suggesting that gigahertz peaked-spectrum sources are as common in the mJy population as they are at Jy levels.