We present the photometric determination of the bright end of the Lyα luminosity function (LF; at LLyα1043.3 erg s-1) within four redshift windows (Δ 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 ∼1000 deg2, with limiting magnitude r ∼ 22. The analysis of high-z Lyα-emitting sources over such a wide area is unprecedented and allows us to select approximately 14 500 hyper-bright (LLyα > 1043.3 erg s-1) Lyα-emitting candidates. We test our selection with two spectroscopic programs at the GTC telescope, which confirm ∼89% of the targets as line-emitting sources, with ∼64% being genuine z ∼ 2.2 quasars (QSOs). We extend the 2.2 z 3.3 Lyα LF for the first time above LLyα ∼ 1044 erg s-1 and down to densities of ∼10-8 Mpc-3. Our results unveil the Schechter exponential decay of the brightest-end of the Lyα LF in great detail, complementing the power-law component of previous determinations at 43.3 Log10(LLyα/erg s-1) 44. We measure φ∗ = (3.33 ± 0.19)×10-6, Log(L∗) = 44.65 ± 0.65, and α = -1.35 ± 0.84 as an average over the probed redshifts. These values are significantly different from the typical Schechter parameters measured for the Lyα LF of high-z star-forming Lyman-α emitters (LAEs). This implies that z > 2 AGNs/QSOs (likely dominant in our samples) are described by a structurally different LF from that used to describe z > 2 star-forming LAEs, namely LQSOs∗ ∼ 100LLAEs∗ and φQSOs∗ ∼ 10-3 φLAEs∗, with the transition between the two LFs happening at LLyα ∼ 1043.5 erg s-1. This supports the scenario in which Lyα-emitting AGNs/QSOs are the most abundant class of z 2 Lyα emitters at LLyα 1043.3 erg s-1. Finally, we suggest that a significant number of these z 2 AGNs/QSOs (∼60% of our samples) are currently misclassified as stars based on their broad-band colours, but are identified for the first time as high-z line-emitters by our narrow-band-based selection.