We report in situ LA-ICP-MS trace element analyses of silicate phases in olivine-bearing chondrules in the Sahara 97096 (EH3) enstatite chondrite. Most olivine and enstatite present rare earth element (REE) patterns comparable to their counterparts in type I chondrules in ordinary chondrites. They thus likely share a similar igneous origin, likely under similar redox conditions. The mesostasis however frequently shows negative Eu and/or Yb (and more rarely Sm) anomalies, evidently out of equilibrium with olivine and enstatite. We suggest that this reflects crystallization of oldhamite during a sulfidation event, already inferred by others, during which the mesostasis was molten, where the complementary positive Eu and Yb anomalies exhibited by oldhamite would have possibly arisen due to a divalent state of these elements. Much of this igneous oldhamite would have been expelled from the chondrules, presumably by inertial acceleration or surface tension effects, and would have contributed to the high abundance of opaque nodules found outside them in EH chondrites. In two chondrules, olivine and enstatite exhibit negatively sloped REE patterns, which may be an extreme manifestation of a general phenomenon (possibly linked to near-liquidus partitioning) underlying the overabundance of light REE observed in most chondrule silicates relative to equilibrium predictions. The silicate phases in one of these two chondrules show complementary Eu, Yb, and Sm anomalies providing direct evidence for the postulated occurrence of the divalent state for these elements at some stage in the formation reservoir of enstatite chondrites. Our work supports the idea that the peculiarities of enstatite chondrites may not require a condensation sequence at high C/O ratios as has long been believed.