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 \pm 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 \lvert \dot { P } / P \lvert of 0.018 yr ^ { -1 } , decreasing from \sim 1340 to \sim 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 .