We report the discovery of a giant Lyα emitter (LAE) with a Spitzer/IRAC counterpart near the reionization epoch at z = 6.595. The giant LAE is found from the extensive 1 deg 2 Subaru narrow-band survey for z = 6.6 LAEs in the Subaru/XMM-Newton Deep Survey (SXDS) field, and subsequently identified by deep spectroscopy of Keck/DEIMOS and Magellan/IMACS. Among our 207 LAE candidates, this LAE is not only the brightest narrow-band object with L(Lyα) = 3.9±0.2×10 43 erg s −1 in our survey volume of 10 6 Mpc 3 , but also a spatially extended Lyα nebula with the largest isophotal area whose major axis is at least ≃ 3 ′′. This object is more likely to be a large Lyα nebula with a size of 17-kpc than to be a strongly-lensed galaxy by a foreground object. Our Keck spectrum with medium-high spectral and spatial resolutions suggests that the velocity width is v FWHM = 251±21 km s −1 , and that the line-center velocity changes by ≃ 60 km s −1 in a 10-kpc range. The stellar mass and star-formation rate are estimated to be 0.9 − 5.0 × 10 10 M ⊙ and > 34 M ⊙ yr −1 , respectively, from the combination of deep optical to infrared images of Subaru, UKIDSS-Ultra Deep Survey, and Spitzer/IRAC. Although the nature of this object is not yet clearly understood, this could be an important object for studying cooling clouds accreting onto a massive halo, or formingmassive galaxies with significant outflows contributing to cosmic reionization and metal enrichment of inter-galactic medium.