Spontaneous symmetry breaking (SSB) is crucial to the occurrence of phase transitions. Once a phase transition occurs, a quantum system presents degenerate eigenstates that lack the symmetry of the Hamiltonian. After crossing the critical point, the system is essentially evolved to a quantum superposition of these eigenstates until decoherence sets in. Despite the fundamental importance and potential applications in quantum technologies, such quantum-mechanical SSB phenomena have not been experimentally explored in many-body systems. We here present an experimental demonstration of the SSB process in the Lipkin-Meshkov-Glick model, governed by the competition between the individual driving and intraqubit interaction. The model is realized in a circuit quantum electrodynamics system, where 6 Xmon qubits are coupled in an all-to-all manner through virtual photon exchange mediated by a resonator. The observed nonclassical correlations among these qubits in the symmetry-breaking region go beyond the conventional description of SSB, shedding new light on phase transitions for quantum many-body systems.