  The photoproduction of vector mesons in $pPb$ collisions at LHC energies is
investigated assuming that the color singlet $t$-channel exchange carries a
large momentum transfer $t$. The rapidity distributions and total cross
sections for the process $Pb \, p \rightarrow Pb \otimes V \otimes \,jet + X$,
with $V = \rho, \, J/\Psi$ and $\otimes$ representing a rapidity gap in the
final state, are estimated considering the non-forward solution of the BFKL
equation at high energy and large -- $t$. A comparison with the predictions
obtained at the Born level also is presented. We predict a large enhancement of
the cross sections associated to the BFKL dynamics in the kinematical range
probed by the LHCb Collaboration. Moreover, our results indicate that the
experimental identification can be feasible at the LHC and that this process
can be used to probe the BFKL dynamics.
