Context. About 120 Be/X-ray binaries (BeXBs) are known in the Small Magellanic Cloud (SMC); about half of them are pulsating with periods from a few to hundreds of seconds. SXP 1323 is one of the longest-period pulsars known in this galaxy. Aims. SXP 1323 is in the field of view of a large set of calibration observations that we analyse systematically, focusing on the time analysis, in search of periodic signals. Methods. We analyse all available X-ray observations of SXP 1323 from Suzaku, XMM-Newton, and Chandra, in the time range from 1999 to the end of 2016. We perform a Lomb-Scargle periodogram search in the band 2.5-10 keV on all observations to detect the neutron star spin period and constrain its long-term evolution. We also perform an orbital period search on the long-term light curve, merging all datasets. Results. We report the discovery of a 26.188+-0.045 d period analysing data from Suzaku, XMM-Newton, and Chandra, which confirms the optical period derived from the Optical Gravitational Lensing Experiment (OGLE) data. If this corresponds to the orbital period, this would be very short with respect to what is expected from the spin/orbital period relationship. We furthermore report on the spin period evolution in the last years. The source is spinning-up with an average rate of Pdot/P of 0.018 yr-1, decreasing from 1340 to 1100 s, in the period from 2006 to the end of 2016, which is also extreme with respect to the other Be/X-ray pulsars. From 2010 to the end of 2014, the pulse period is not clearly detectable, although the source was still bright. Conclusions. SXP 1323 is a peculiar BeXB due to its long pulse period, rapid spin-up for several years, and short orbital period. A continuous monitoring of the source in the next years is necessary to establish the long-term behaviour of the spin period.
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