We present the photometric determination of the bright-end of the Ly{alpha} luminosity function (at L_Ly{alpha}_>~10^43.5^erg/s) within four redshifts windows ({Delta}z<0.16) in the interval 2.2<~z<~3.3. Our work is based on the Javalambre Photometric Local Universe Survey (J-PLUS) first data-release, which provides multiple narrow-band measurements over ~1000deg^2^, with limiting magnitude ~22. Theanalysis of high-z Ly{alpha}-emitting sources over such a wide area is unprecedented, and allows to select a total of ~14500 hyper-bright(L_Ly{alpha}_>10^43.3^erg/s) Ly{alpha}-emitting candidates. We test our selection with two spectroscopic follow-up programs at the GTC telescope,which confirm as line-emitting sources ~89% of the targets, with ~64% being genuine z~2.2 QSOs. We extend the 2.2<~z<~3.3 Ly{alpha} luminosity function for the first time above L_Ly{alpha}_~10^44^erg/s and down to densities of ~10^-8^Mpc^-3^. Our results unveil with high detail the Schechter exponential-decay of the brightest-end of the Ly{alpha} LF, complementing the power-law component of previous LF determinations at 43.3<~Log10(L_Ly{alpha}_/(erg/s))<~44. We measure {PHI}*=(3.33+/-0.19)x10^-6^, Log(L*)=44.65+/-0.65 and {alpha}=-1.35+/-0.84 as an average over the redshifts we probe. These values are significantly different than the typical Schechter parameters measured for the Ly{alpha} LF of high-z star-forming LAEs. This suggests that z>2 AGN/QSOs (likely dominant in our samples) are described by a structurally different LF than z>2 star-forming LAEs, namely with L*_QSOs_~100 L*_LAEs_ and {PHI}*_QSOs_~10^-3^{PHI}*_LAEs_. Finally, our method identifies very efficiently as high-z line-emitters sources without previous spectroscopic confirmation, currently classified as stars (~2000 objects in each redshift bin, on average). Assuming a large predominance of Ly{alpha}-emitting AGN/QSOs in our samples, this supports the scenario by which these are the most abundant class of z>~2 Ly{alpha} emitters at L_Ly{alpha}_>~10^43.3^erg/s.