  We present a spectroscopic redshift catalogue of the SMACS J0723.3$-$7327
field ("Webb's First Deep Field") obtained from JWST/NIRISS grism spectroscopy
and supplemented with JWST/NIRSpec and VLT/MUSE redshifts. The catalogue
contains a total of 190 sources with secure spectroscopic redshifts, including
156 NIRISS grism redshifts, 123 of which are for sources whose redshifts were
previously unknown. These new grism redshifts are secured with two or more
spectroscopic features (64 sources), or with a single spectral feature whose
identity is secured from the object's nine-band photometric redshift (59
sources). These are complemented with 17 NIRSpec and 48 MUSE redshifts,
including six new NIRSpec redshifts identified in this work. In addition to the
$z_{\rm cl}=0.39$ cluster galaxy redshifts (for which we provide $\sim$40 new
NIRISS absorption-line redshifts), we also find three prominent galaxy
overdensities at higher redshifts - at $z=1.1$, $z=1.4$, and $z=2.0$ - that
were until now not seen in the JWST/NIRSpec and VLT/MUSE data. The paper
describes the characteristics of our spectroscopic redshift sample and the
methodology we have employed to obtain it. Our redshift catalogue is made
available to the community at https://niriss.github.io/smacs0723.
