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ABSTRACT Recent searches for the hosts of z ∼ 4 damped Ly α absorbers (DLAs) have detected bright galaxies at distances of tens of kpc from the DLA. Using the FIRE-2 cosmological zoom simulations, we argue that these relatively large distances are due to a predominantly cool and neutral inner circumgalactic medium (CGM) surrounding high-redshift galaxies. The inner CGM is cool because of the short cooling time of hot gas in $${\lesssim}10^{12}\, {\rm M_{\odot }}$$ haloes, which implies that accretion and feedback energy are radiated quickly, while it is neutral due to high volume densities and column densities at high redshift that shield cool gas from photoionization. Our analysis predicts large DLA covering factors ($${\gtrsim}50{{\ \rm per\ cent}}$$) out to impact parameters ∼0.3[(1 + z)/5]3/2Rvir from the central galaxies at z ≳ 1, equivalent to a proper distance of $${\sim}21\, M_{12}^{1/3} \left(\left(1+z\right)/5\right)^{1/2}\, {\rm kpc}$$ (Rvir and M12 are the halo virial radius and mass in units of $$10^{12}\, {\rm M_{\odot }}$$, respectively). This implies that DLA covering factors at z ∼ 4 may be comparable to unity out to a distance ∼10 times larger than stellar half-mass radii. A predominantly neutral inner CGM in the early universe suggests that its mass and metallicity can be directly constrained by absorption surveys, without resorting to the large ionization corrections as required for ionized CGM.