: The first measurement of the cross section for incoherent photonuclear production of J/ψ vector mesons as a function of the Mandelstam |t| variable is presented. The measurement was carried out with the ALICE detector at midrapidity, |y|<0.8, using ultraperipheral collisions of Pb nuclei at a center-of-mass energy per nucleon pair of sqrt[s_{NN}]=5.02 TeV. This rapidity interval corresponds to a Bjorken-x range (0.3-1.4)×10^{-3}. Cross sections are given in five |t| intervals in the range 0.04<|t|<1 GeV^{2} and compared to the predictions by different models. Models that ignore quantum fluctuations of the gluon density in the colliding hadron predict a |t| dependence of the cross section much steeper than in data. The inclusion of such fluctuations in the same models provides a better description of the data.
