  We present a reduction and analysis of the \textit{James Webb Space
Telescope} (JWST) SMACS~0723 field using new post-launch calibrations to
conduct a search for ultra-high-redshift galaxies ($z > 9$) present within the
Epoch of Reionisation. We conduct this search by modelling photometric
redshifts in several ways for all sources and by applying conservative
magnitude cuts ($m_{\rm F200W} < 28$) to identify strong Lyman breaks greater
than 1 magnitude. We find four $z > 9$ candidate galaxies which have not
previously been identified, with one object at $z = 11.5$, and another which is
possibly a close pair of galaxies. We measure redshifts for candidate galaxies
from other studies and find the recovery rate to be only 23 per cent, with many
being assigned lower redshift, dusty solutions in our work. Most of our $z > 9$
sample show evidence for Balmer-breaks, or extreme emission lines from H$\beta$
and [OIII], demonstrating that the stellar populations could be advanced in age
or very young depending on the cause of the F444W excess. We discuss the
resolved structures of these early galaxies and find that the S\'{e}rsic
indices reveal a mixture of light concentration levels, but that the sizes of
all our systems are exceptionally small ($< 0.5$~kpc). These systems have
stellar masses M$_{*} \sim 10^{9.0}$ M$_{\odot}$, with our $z \sim 11.5$
candidate a dwarf galaxy with a stellar mass M$_{*} \sim 10^{7.8}$ --
$10^{8.2}$ M$_{\odot}$. These candidate ultra high-redshift galaxies are
excellent targets for future NIRSpec observations aimed to better understand
their physical nature.
