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We simulate, using nonperturbative methods, the real-time dynamics of small bubbles of "false vacuum" in a quantum spin chain near criticality, where the low-energy physics is described by a relativistic (1+1)-dimensional quantum field theory. We consider bubbles whose walls are kink and antikink quasiparticle excitations, so that wall collisions are kink-antikink scattering events. To construct these bubbles in the presence of strong correlations, we extend a recently proposed matrix product state (MPS) ansatz for quasiparticle wavepackets. We simulate dynamics within a window of about 1000 spins embedded in an infinite chain at energies of up to about 5 times the mass gap. By choosing the wavepacket width and the bubble size appropriately, we avoid strong lattice effects and observe relativistic kink-antikink collisions. We use the MPS quasiparticle ansatz to detect scattering outcomes. (i) In the Ising model, with transverse and longitudinal fields, we do not observe particle production despite nonintegrability (supporting recent observations of nonthermalizing states in this model). (ii) Switching on an additional interaction, we see production of confined and unconfined particle pairs. We characterize the amount of entanglement generated as a function of energy and time and conclude that our classical simulation methods will ultimately fail as these increase. We anticipate that kink-antikink scattering in 1+1 dimensions will be an instructive benchmark problem for future quantum computers and analog quantum simulators.