Deep multipass cutting of bidirectional and unidirectional carbon fiber reinforced plastics (CFRP) with picosecond laser pulses was investigated in different static atmospheres as well as with the assistance of an oxygen or nitrogen gas flow. The ablation rate was determined as a function of the kerf depth and the resulting heat affected zone was measured. An assisting oxygen gas flow is found to significantly increase the cutting productivity, but only in deep kerfs where the diminished evaporative ablation due to the reduced laser fluence reaching the bottom of the kerf does not dominate the contribution of reactive etching anymore. Oxygen-supported cutting was shown to also solve the problem that occurs when cutting the CFRP parallel to the fiber orientation where a strong deformation and widening of the kerf, which temporarily slows down the process speed, is revealed to be typical for processing in standard air atmospheres.