We report the discovery of a giant Ly \alpha 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 ( { Ly } \alpha ) = 3.9 \pm 0.2 \times 10 ^ { 43 } erg s ^ { -1 } in our survey volume of 10 ^ { 6 } Mpc ^ { 3 } , but also a spatially extended Ly \alpha nebula with the largest isophotal area whose major axis is at least \simeq 3 ^ { \prime \prime } . This object is more likely to be a large Ly \alpha nebula with a size of \gtrsim 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 \pm 21 km s ^ { -1 } , and that the line-center velocity changes by \simeq 60 km s ^ { -1 } in a 10 -kpc range . The stellar mass and star-formation rate are estimated to be 0.9 - 5.0 \times 10 ^ { 10 } M _ { \odot } and > 34 M _ { \odot } 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 forming-massive galaxies with significant outflows contributing to cosmic reionization and metal enrichment of inter-galactic medium .