  We apply the Lyman absorption signature to search for galaxies at redshifts z
\~ 6 - 17 using optical and infrared images of the Hubble Deep Field. The
infrared images are sensitive to a point source 5 sigma detection threshold of
AB(22,000) = 23.8, which adopting plausible assumptions to relate rest-frame
ultraviolet flux densities to unobscured star formation rates is easily
sufficient to detect the star formation rates expected for massive elliptical
galaxy formation to quite high redshifts. For q_0 = 0.5, the infrared images
are sensitive to an unobscured star formation rate of 100 h^-2 solar masses per
year to redshifts as large as z = 17, and for q_0 = 0, the infrared images are
sensitive to an unobscured star formation rate of 300 h^-2 solar masses per
year to redshifts as large as z = 14. The primary result of the analysis is
that only one extremely high redshift galaxy candidate is identified at the 5
sigma level of significance (and four at the 4 sigma level). This implies a
strict upper limit to the surface density of extremely high redshift galaxies
of < 1.5 arcmin^-2 to a limiting magnitude threshold AB(22,000) = 23.8. This
also implies a strict upper limit to the volume density of extremely high
redshift galaxies if (and only if) such galaxies are not highly obscured by
dust.
