We study the global stability of non-axisymmetric p modes (also called inertial-acoustic modes) trapped in the innermost regions of accretion discs around black holes. We show that the lowest-order (highest-frequency) p modes, with frequencies ω = (0.5-0.7) mΩISCO [where m = 1, 2, 3, ... is the azimuthal wavenumber, ΩISCO is the disc rotation frequency at the Innermost Stable Circular Orbit (ISCO)], can be overstable due to general relativistic effects, according to which the radial epicyclic frequency κ is a non-monotonic function of radius near the black hole. The mode is trapped inside the corotation resonance radius rc (where the wave pattern rotation speed ω/m equals the disc rotation rate Ω) and carries a negative energy. The mode growth arises primarily from wave absorption at the corotation resonance, and the sign of the wave absorption depends on the gradient of the disc vortensity, ζ = κ2/(2ΩΣ) (where Σ is the surface density). When the mode frequency ω is sufficiently high, such that dζ/r > 0 at rc, positive wave energy is absorbed at the corotation, leading to the growth of mode amplitude. The mode growth is further enhanced by wave transmission beyond the corotation barrier. We also study how the rapid radial inflow at the inner edge of the disc affects the mode trapping and growth. Our analysis of the behaviour of the fluid perturbations in the transonic flow near the ISCO indicates that, while the inflow tends to damp the mode, the damping effect is sufficiently small under some conditions (e.g. when the disc density decreases rapidly with decreasing radius at the sonic point) so that net mode growth can still be achieved. We further clarify the role of the Rossby wave instability and show that it does not operate for black hole accretion discs with smooth-varying vortensity profiles. Overstable non-axisymmetric p modes driven by the corotational instability provide a plausible explanation for the high-frequency (>~100 Hz) quasi-periodic oscillations observed from a number of black-hole X-ray binaries in the very high state. The absence of high-frequency quasi-periodic oscillations in the soft (thermal) state may result from mode damping due to the radial infall at the ISCO.