  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 $S_{1.2mm}=14\pm5\mu$Jy. This implies a dust-obscured
star formation rate (SFR) of $(3\pm1)$ 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.
