  The ~1 Ms Chandra Deep Field North observation is used to study the extended
X-ray sources in the region surrounding the Hubble Deep Field North (HDF-N),
yielding the most sensitive probe of extended X-ray emission at cosmological
distances to date. A total of six such sources are detected, the majority of
which align with small numbers of optically bright galaxies. Their angular
sizes, band ratios, and X-ray luminosities -- assuming they lie at the same
distances as the galaxies coincident with the X-ray emission -- are generally
consistent with the properties found for nearby groups of galaxies. One source
is notably different and is likely to be a poor-to-moderate X-ray cluster at
high redshift (i.e., z > 0.7). We are also able to place strong constraints on
the optically detected cluster of galaxies ClG 1236+6215 at z=0.85 and the
wide-angle-tail radio galaxy VLA J123725.7+621128 at z~1-2. With rest-frame
0.5--2.0 keV X-ray luminosities of <(3-15)e42 ergs s^{-1}, the environments of
both sources are either likely to have a significant deficit of hot
intra-cluster gas compared to local clusters of galaxies, or they are X-ray
groups. We find the surface density of extended X-ray sources in this
observation to be 167 (+97,-67) deg^{-2} at a limiting soft-band flux of
approximately 3e-16 ergs s^{-1} cm^{-2}. No evolution in the X-ray luminosity
function of clusters is needed to explain this value. (Abridged)
