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$_{rm [CII]}sim$(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 $S_{1.2mm}=14pm5mu$Jy. This implies a dust-obscured star formation rate (SFR) of $(3pm1)$ 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 $L_{rm [CII]}$ 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.
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