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Recently, the mean free path of ionizing photons in the $z = 6$ intergalactic medium (IGM) was measured to be very short, presenting a challenge to existing reionization models. At face value, the measurement can be interpreted as evidence that the IGM clumps on scales $M\lesssim 10^8$ M$_\odot$, a key but largely untested prediction of the cold dark matter (CDM) paradigm. Motivated by this possibility, we study the role that the underlying dark matter cosmology plays in setting the $z > 5$ mean free path. We use two classes of models to contrast against the standard CDM prediction: (1) thermal relic warm dark matter (WDM), representing models with suppressed small-scale power; (2) an ultralight axion exhibiting a white noise-like power enhancement. Differences in the mean free path between the WDM and CDM models are subdued by pressure smoothing and the possible contribution of neutral islands to the IGM opacity. For example, comparing late reionization scenarios with a fixed volume-weighted mean neutral fraction of $20\%$ at $z=6$, the mean free path is $19~(45)~\%$ longer in a WDM model with $m_x = 3~(1)$ keV. The enhanced power in the axion-like model produces better agreement with the short mean free path measured at $z = 6$. However, drawing robust conclusions about cosmology is hampered by large uncertainties in the reionization process, extragalactic ionizing background, and thermal history of the Universe. This work highlights some key open questions about the IGM opacity during reionization.