We present a survey for low-ionization metal absorption line systems towards 17 QSOs at redshifts z_em=5.8-6.4. Nine of our objects were observed at high resolution with either Keck/HIRES or Magellan/MIKE, and the remainder at moderate resolution with Keck/ESI. The survey spans 5.3 < z_abs < 6.4 and has a pathlength interval Delta X=39.5, or Delta z=8.0. In total we detect ten systems, five of which are new discoveries. The line-of-sight number density is consistent with the combined number density at z~3 of DLAs and sub-DLAs, which comprise the main population of low-ionization systems at lower redshifts. This apparent lack of evolution may occur because low ionization systems are hosted by lower-mass halos at higher redshifts, or because the mean cross section of low-ionization gas at a given halo mass increases with redshift due to the higher densities and lower ionizing background. The roughly constant number density notably contrasts with the sharp decline at z > 5.3 in the number density of highly-ionized systems traced by C IV. The low-ionization systems at z~6 span a similar range of velocity widths as lower-redshift sub-DLAs but have significantly weaker lines at a given width. This implies that the mass-metallicity relation of the host galaxies evolves towards lower metallicities at higher redshifts. These systems lack strong Si IV and C IV, which are common among lower-redshift DLAs and sub-DLAs. This is consistent, however, with a similar decrease in the metallicity of the low- and high-ionization phases, and does not necessarily indicate a lack of nearby, highly-ionized gas. The high number density of low-ionization systems at z~6 suggests that we may be detecting galaxies below the current limits of i-dropout and Ly-alpha emission galaxy surveys. These systems may therefore be the first direct probes of the `typical galaxies responsible for hydrogen reionization.
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