Past observations of QSO 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 PdBI follow-up observations of the [ C ii ] 158 \mu m emission line and FIR continuum in the host galaxy of SDSS J1148+5251 , a luminous QSO at redshift 6.4189 .
We find that a large fraction of the gas traced by [ C ii ] is at high velocities , up to \sim 1400 km s ^ { -1 } relative to the systemic velocity , confirming the presence of a major outflow indicated by previous observations .
The outflow has a complex morphology and reaches a maximum projected radius of \simeq 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 \sim 20 Myr .
Surprisingly , we discover that also the quiescent gas at systemic velocity is extremely extended .
More specifically , we find that , while 30 % of the [ C ii ] within v \in ( -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 FWHM size of 17.4 \pm 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 \pm 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 \sim 30 kpc in the host galaxy of SDSS J1148+5251 .
A large fraction of extended [ C ii ] is likely associated with star formation occurring on large scales , but the [ C ii ] source extends well beyond the FIR continuum , and additional multi-wavelengths observations are needed in order to clarify the origin of this most extended [ C ii ] .