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We investigate the prospects of the diffractive production of $J/\psi$ mesonsat large momentum transfer $|t|$ at the future Electron Ion Collider inelectron-proton collisions. In particular, we focus on the measurements of therapidity gap size. The model used for the calculations is based on thediffractive exchange of the Balitsky-Fadin-Kuraev-Lipatov perturbative Pomeron.Calculations for the cross section and the estimates for the rates assumingintegrated luminosity of $10 \, \rm fb^{-1}$ are provided. Two experimentalstrategies were considered. First, measuring the rapidity gap size directly, byobserving the activity in the forward part of the central detector, and secondby putting a lower limit on the rapidity gap size in the case when the detectorcannot measure forward activity. We find that, it is possible to measure at theEIC the dependence of the cross section on rapidity gap interval up to fourunits in rapidity. This should allow to measure the change of the cross sectionby a factor 1.6 expected due to the BFKL exchange. This is possible with thepresent setup of the detector which projects the coverage up to 3.5 units ofrapidity. We conclude however, that the extension of the detector up to higherrapidity, for example to 4.5 would be desirable and provide even better leverarm for testing rapidity gap physics at the EIC.

High $|t|$ diffractive vector meson production at the EIC