We study six luminous Lyman-alpha emitters (LAEs) with very blue rest-frame UV continua at $5.7le z le 6.6$. These LAEs have previous HST and Spitzer IRAC observations. Combining our newly acquired HST images, we find that their UV-continuum slopes $beta$ are in a range of $-3.4le beta le -2.6$. Unlike previous, tentative detections of $beta simeq -3$ in photometrically selected, low-luminosity galaxies, our LAEs are spectroscopically confirmed and luminous ($M_{rm UV}<-20$ mag). We model their broadband spectral energy distributions (SEDs), and find that two $betasimeq-2.6pm0.2$ galaxies can be well fitted with young and dust-free stellar populations. However, it becomes increasingly difficult to fit bluer galaxies. We explore further interpretations by including non-zero LyC escape fraction $f_{rm esc}$, very low metallicities, and/or AGN contributions. Assuming $f_{rm esc}simeq0.2$, we achieve the bluest slopes $betasimeq-2.7$ when nebular emission is considered. This can nearly explain the SEDs of two galaxies with $betasimeq-2.8$ and --2.9 ($sigma_{beta}=0.15$). Larger $f_{rm esc}$ values and very low metallicities are not favored by the strong nebular line emission (evidenced by the IRAC flux) or the observed (IRAC 1 - IRAC 2) color. Finally, we find that the $betasimeq-2.9$ galaxy can potentially be well explained by the combination of a very young population with a high $f_{rm esc}$ ($ge0.5$) and an old, dusty population. We are not able to produce two $beta simeq -3.4 pm0.4$ galaxies. Future deep spectroscopic observations are needed to fully understand these galaxies.
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