In contrast to interband excitons in undoped quantum wells, doped quantum wells do not display sharp resonances due to excitonic bound states. In these systems the effective Coulomb interaction between electrons and holes typically only leads to a depolarization shift of the single-electron intersubband transitions. Non-perturbative light-matter interaction in solid-state devices has been investigated as a pathway to tune optoelectronic properties of materials. A recent theoretical work [Cortese et al., Optica 6, 354 (2019)] predicted that, when the doped quantum wells are embedded in a photonic cavity, emission-reabsorption processes of cavity photons can generate an effective attractive interaction which binds electrons and holes together, leading to the creation of an intraband bound exciton. Spectroscopically, this bound state manifests itself as a novel discrete resonance which appears below the ionisation threshold only when the coupling between light and matter is increased above a critical value. Here we report the first experimental observation of such a bound state using doped GaAs/AlGaAs quantum wells embedded in metal-metal resonators whose confinement is high enough to permit operation in strong coupling. Our result provides the first evidence of bound states of charged particles kept together not by Coulomb interaction, but by the exchange of transverse photons. Light-matter coupling can thus be used as a novel tool in quantum material engineering, tuning electronic properties of semiconductor heterostructures beyond those permitted by mere crystal structures, with direct applications to mid-infrared optoelectronics.