ABSTRACT We study the effects of cosmic rays (CRs) on outflows from star-forming galaxies in the circum and intergalactic medium (CGM/IGM), in high-resolution, fully cosmological FIRE-2 simulations (accounting for mechanical and radiative stellar feedback, magnetic fields, anisotropic conduction/viscosity/CR diffusion and streaming, and CR losses). We showed previously that massive ($M_{\rm halo}\gtrsim 10^{11}\, \mathrm{M}_{⊙ }$), low-redshift (z ≲ 1–2) haloes can have CR pressure dominate over thermal CGM pressure and balance gravity, giving rise to a cooler CGM with an equilibrium density profile. This dramatically alters outflows. Absent CRs, high gas thermal pressure in massive haloes ‘traps’ galactic outflows near the disc, so they recycle. With CRs injected in supernovae as modelled here, the low-pressure halo allows ‘escape’ and CR pressure gradients continuously accelerate this material well into the IGM in ‘fast’ outflows, while lower-density gas at large radii is accelerated in situ into ‘slow’ outflows that extend to >Mpc scales. CGM/IGM outflow morphologies are radically altered: they become mostly volume-filling (with inflow in a thin mid-plane layer) and coherently biconical from the disc to >Mpc. The CR-driven outflows are primarily cool ($T∼ \! 10^{5}\,$ K) and low velocity. All of these effects weaken and eventually vanish at lower halo masses ($\lesssim 10^{11}\, \mathrm{M}_{⊙ }$) or higher redshifts (z ≳ 1–2), reflecting the ratio of CR to thermal + gravitational pressure in the outer halo. We present a simple analytical model that explains all of the above phenomena. We caution that these predictions may depend on uncertain CR transport physics.