Context. A significant fraction of the molecular gas in star-forming regions is irradiated by stellar UV photons. In these environments, the electron density (n_e) plays a critical role in the gas dynamics, chemistry, and collisional excitation of certain molecules. Aims. We determine n_e in the prototypical strongly irradiated photodissociation region (PDR), the Orion Bar, from the detection of new millimeter-wave carbon recombination lines (mmCRLs) and existing far-IR [¹³C II] hyperfine line observations. Methods. We detect 12 mmCRLs (including α, β, and γ transitions) observed with the IRAM 30 m telescope, at ∼25″ angular resolution, toward the H/H₂ dissociation front (DF) of the Bar. We also present a mmCRL emission cut across the PDR. Results. These lines trace the C⁺/C/CO gas transition layer. As the much lower frequency carbon radio recombination lines, mmCRLs arise from neutral PDR gas and not from ionized gas in the adjacent H II region. This is readily seen from their narrow line profiles (Δv = 2.6 ± 0.4 km s⁻¹) and line peak velocities (v_LSR = +10.7 ± 0.2 km s⁻¹). Optically thin [¹³C II] hyperfine lines and molecular lines – emitted close to the DF by trace species such as reactive ions CO⁺ and HOC⁺ – show the same line profiles. We use non-LTE excitation models of [¹³C II] and mmCRLs and derive n_e = 60–100 cm⁻³ and T_e = 500–600 K toward the DF. Conclusions. The inferred electron densities are high, up to an order of magnitude higher than previously thought. They provide a lower limit to the gas thermal pressure at the PDR edge without using molecular tracers. We obtain P_th ≥ (2−4) × 10⁸ cm⁻³ K assuming that the electron abundance is equal to or lower than the gas-phase elemental abundance of carbon. Such elevated thermal pressures leave little room for magnetic pressure support and agree with a scenario in which the PDR photoevaporates.