We report the discovery of phase shifts between X-ray pulses at different energies in the newly discovered millisecond (ms) X-ray pulsar SAX J1808.4-3658. The results show that low-energy pulses lag high-energy pulses by as much as $sim$0.2 ms (or $sim$8% of the pulse period). The measurements were made in two different ways: (1) computing cross power spectra between different energy bands, and (2) cross-correlating the folded pulse profiles in different energy bands; consistent results were obtained. We speculate that the observed soft lags might be related to the lateral expansion and subsequent cooling of a ``hot spot on the neutron star surface in which the pulsed X-ray emission originates. Also presented is the possibility of producing soft lags via Compton down scattering of hard X-ray photons from the hot spot in the cool surrounding atmosphere. We will discuss possible X-ray production mechanisms for SAX J1808.4-3658 and constraints on the emission environment, based on the observed soft lags, pulse profiles, and energy spectrum.
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