  We describe new ultra-deep James Webb Space Telescope (JWST) NIRSpec PRISM
and grating spectra for the galaxies JADES-GS-z11-0 ($z_{\mathrm{spec}} =
11.122^{+0.005}_{-0.003}$) and JADES-GS-z13-0 ($z_{\mathrm{spec}} =
13.20^{+0.03}_{-0.04}$), the most distant spectroscopically-confirmed galaxy
discovered in the first year of JWST observations. The extraordinary depth of
these observations (75 hours and 56 hours, respectively) provides a unique
opportunity to explore the redshifts, stellar properties, UV magnitudes, and
slopes for these two sources. For JADES-GS-z11-0, we find evidence for multiple
emission lines, including [OII]3726,3729 and [NeIII]3869, resulting in a
spectroscopic redshift we determine with 94% confidence. At this spectroscopic
redshift, the Lyman-$\alpha$ break in JADES-GS-z11-0 can be fit with a damped
Lyman-$\alpha$ absorber with $\log{(N_\mathrm{HI}/\mathrm{cm}^{-2})} =
22.42^{+0.093}_{-0.120}$. We present stringent upper limits on the emission
line fluxes and line equivalent widths for JADES-GS-z13-0. These results
demonstrate how neutral hydrogen fraction and Lyman-damping wings may impact
the recovery of spectroscopic redshifts for sources like these, providing
insight into the overprediction of the photometric redshifts seen for distant
galaxies observed with JWST. In addition, we analyze updated NIRCam photometry
to calculate the morphological properties of these resolved sources, and find a
secondary source $0.3^{\prime\prime}$ south of JADES-GS-z11-0 at a similar
photometric redshift, hinting at how galaxies grow through interactions in the
early Universe.
