We investigate the kinetics of a polymer collapse due to the formation of irreversible cross-links between its monomers. Using the contact probability P(s) as a scale-dependent order parameter depending on the chemical distance s, our simulations show the emergence of a cooperative pearling instability. Namely, the polymer undergoes a sharp conformational transition to a set of absorbing states characterized by a length scale ξ corresponding to the mean pearl size. This length and the transition time depend on the polymer equilibrium dynamics and the cross-linking rate. We confirm experimentally this transition using a DNA conformation capture experiment in yeast.