  A supersolid is a counter-intuitive phase of matter where its constituent
particles are arranged into a crystalline structure, yet they are free to flow
without friction. This requires the particles to share a global macroscopic
phase while being able to reduce their total energy by spontaneous, spatial
self-organisation. This exotic state of matter has been achieved in different
systems using Bose-Einstein condensates coupled to cavities, possessing
spin-orbit coupling, or dipolar interactions. Here we provide experimental
evidence of a new implementation of the supersolid phase in a novel
non-equilibrium context based on exciton-polaritons condensed in a
topologically non-trivial, bound-in-the-continuum state with exceptionally low
losses. We measure the density modulation of the polaritonic state indicating
the breaking of translational symmetry with a remarkable precision of a few
parts in a thousand. Direct access to the phase of the wavefunction allows us
to additionally measure the local coherence of the superfluid component. We
demonstrate the potential of our synthetic photonic material to host phonon
dynamics and a multimode excitation spectrum.
