Past observations of quasar host galaxies at z> 6 have found cold gas and star formation on compact scales of a few kiloparsecs. We present new high sensitivity IRAM Plateau de Bure Interferometer follow-up observations of the C ii 158 μm emission line and far-infrared (FIR) continuum in the host galaxy of SDSS J1148+5251, a luminous quasar at redshift 6.4189. We find that a large portion of the gas traced by C ii is at high velocities, up to ~1400 km s-1relative to the systemic velocity, confirming the presence of a major outflow as indicated by previous observations. The outflow has a complex morphology and reaches a maximum projected radius of ≃30 kpc. The extreme spatial extent of the outflow allows us, for the first time in an external galaxy, to estimate mass-loss rate, kinetic power, and momentum rate of the outflow as a function of the projected distance from the nucleus and the dynamical time scale. These trends reveal multiple outflow events during the past 100 Myr, although the bulk of the mass, energy, and momentum appear to have been released more recently within the past ~20 Myr. Surprisingly, we discover that the quiescent gas at systemic velocity is also extremely extended. More specifically, we find that, while 30% of the C ii within v ∈(−200, 200) km s-1 traces a compact component that is not resolved by our observations, 70% of the C ii emission in this velocity range is extended with a projected full width at half maximum (FWHM) size of 17.4 ± 1.4 kpc. We detect FIR continuum emission associated with both the compact and the extended C ii components, although the extended FIR emission has a FWHM of 11 ± 3 kpc, thus smaller than the extended C ii source. Overall, our results indicate that the cold gas traced by C ii is distributed up to r ~ 30 kpc in the host galaxy of SDSS J1148+5251. A large amount of extended C ii is likely to be associated with star formation occurring on large scales, but the C ii source extends well beyond the FIR continuum, and additional multi-wavelength observations are needed in order to clarify the origin of this very extended C ii .