We present two-dimensional spectroscopy of the 125×125 pc2 circumnuclear region of M31, obtained using two optical fiber systems, INTEGRAL and 2D-FIS, installed at the 4.2 m William Herschel Telescope at the Roque de los Muchachos Observatory on the island of La Palma, Spain. We present continuum and emission-line maps, emission-line quotients, and stellar and ionized gas velocity fields obtained under subarcsecond seeing conditions. We have discovered Hβ, [O III] λλ4959, 5007, Hα, and [N II] λλ6548, 6584 emission in the inner 40×40 pc2 circumnuclear region. The intensity maps obtained from the emission lines show several isolated systems of ionized gas (clouds) a few parsecs in size. The kinematics of the clouds is decoupled from the stellar rotation, and strong radial motions are likely to be present. We have also discussed the ionization mechanisms present in the clouds. Although the emission-line quotients are compatible with ionization produced by post-asymptotic giant branch stars, the estimated ionization fluxes exceed what is expected according to the distribution of planetary nebulae in the bulge of M31. Comparison of the line quotients and spectra found in M31 with those found in the extended narrow-line region of the Seyfert galaxy NGC 4151 indicates that we could be in presence of ultralow-intensity nuclear activity. These results would be compatible with the existence of a ``dead quasar'' in the nucleus of M31 and underline the hybrid character of the LINER phenomenon.