  We present a robust sample of very high-redshift galaxy candidates from the
first epoch of {\it JWST}/NIRCam imaging from the Next Generation Extragalactic
Exploratory Deep (NGDEEP) Survey. The NGDEEP NIRCam imaging in the Hubble Ultra
Deep Field Parallel Field 2 (HUDF-Par2) reaches $m=30.4$ (5$\sigma$,
point-source) in F277W, making it the deepest public {\it JWST} GO imaging
dataset to date. We describe our detailed data reduction process of the
six-filter broad-band {\it JWST}/NIRCam imaging, incorporating custom
corrections for systematic effects to produce high-quality calibrated images.
Using robust photometric redshift selection criteria, we identify a sample of
38 $z \gtrsim 9$ galaxy candidates. These objects span a redshift range of
$z=8.5-15.8$, and apparent magnitudes of $m_\mathrm{F277W} = 27-30.5$ AB mag,
reaching $\sim 1.5$ mag deeper than previous public {\it JWST} imaging surveys.
We calculate the rest-frame ultraviolet (UV) luminosity function at $z \sim$ 9
and 11, and present a new measurement of the luminosity function faint-end
slope at $z \sim 11$. There is no significant evolution in the faint-end slope
and number density from $z=9$ to 11. Comparing our results with theoretical
predictions, we find that some models produce better agreement at the faint end
than the bright end. These results will help to constrain how stellar feedback
impacts star formation at these early epochs.
