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Abstract The Atacama Large Millimeter/submillimeter Array (ALMA) Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS) Band 6 scan (212–272 GHz) covers potential [C ii ] emission in galaxies at 6 ≤ z ≤ 8 throughout a 2.9 arcmin 2 area. By selecting on known Ly α emitters (LAEs) and photometric dropout galaxies in the field, we perform targeted searches down to a 5 σ [C ii ] luminosity depth L [C II ] ∼ 2.0 × 10 8 L ⊙ , corresponding roughly to star formation rates (SFRs) of 10–20 M ⊙ yr −1 when applying a locally calibrated conversion for star-forming galaxies, yielding zero detections. While the majority of galaxies in this sample are characterized by lower SFRs, the resulting upper limits on [C ii ] luminosity in these sources are consistent with the current literature sample of targeted ALMA observations of z = 6–7 LAEs and Lyman-break galaxies (LBGs), as well as the locally calibrated relations between L [C ii ] and SFR—with the exception of a single [C ii ]-deficient, UV-luminous LBG. We also perform a blind search for [C ii ]-bright galaxies that may have been missed by optical selections, resulting in an upper limit on the cumulative number density of [C ii ] sources with L [C II ] > 2.0 × 10 8 L ⊙ (5 σ ) to be less than 1.8 × 10 −4 Mpc −3 (90% confidence level). At this luminosity depth and volume coverage, we present an observed evolution of the [C ii ] luminosity function from z = 6–8 to z ∼ 0 by comparing the ASPECS measurement to literature results at lower redshift.