The first large catalogue of spectroscopic redshifts in Webb's First Deep Field, SMACS J0723.3$-$7327

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.