We present a search for [ CII ] line and dust continuum emission from optical dropout galaxies at z > 6 using ASPECS , our ALMA Spectroscopic Survey in the Hubble Ultra–Deep Field ( UDF ) . Our observations , which cover the frequency range 212 - 272 GHz , encompass approximately the range 6 < z < 8 for [ CII ] line emission and reach a limiting luminosity of L _ { [ CII ] } \sim ( 1.6-2.5 ) \times 10 ^ { 8 } L _ { \odot } . We identify fourteen [ CII ] line emitting candidates in this redshift range with significances > 4.5 \sigma , two of which correspond to blind detections with no optical counterparts . At this significance level , our statistical analysis shows that about 60 % of our candidates are expected to be spurious . For one of our blindly selected [ CII ] line candidates , we tentatively detect the CO ( 6-5 ) line in our parallel 3-mm line scan . None of the line candidates are individually detected in the 1.2 mm continuum . A stack of all [ CII ] candidates results in a tentative detection with S _ { 1.2 mm } = 14 \pm 5 \mu Jy . This implies a dust–obscured star formation rate ( SFR ) of ( 3 \pm 1 ) M _ { \odot } yr ^ { -1 } . We find that the two highest–SFR objects have candidate [ CII ] lines with luminosities that are consistent with the low–redshift L _ { [ CII ] } vs. SFR relation . The other candidates have significantly higher [ CII ] luminosities than expected from their UV–based SFR . At the current sensitivity it is unclear whether the majority of these sources are intrinsically bright [ CII ] emitters , or spurious sources . If only one of our line candidates was real ( a scenario greatly favored by our statistical analysis ) , we find a source density for [ CII ] emitters at 6 < z < 8 that is significantly higher than predicted by current models and some extrapolations from galaxies in the local universe .