When the Fermi level matches the Dirac point in graphene, the reduced charge screening can dramatically enhance electron-electron (e-e) scattering to produce a strongly interacting Dirac liquid. While the dominance of e-e scattering already leads to novel behaviors, such as electron hydrodynamic flow, further exotic phenomena have been predicted to arise specifically from the unique kinematics of e-e scattering in massless Dirac systems. Here, we use optoelectronic probes, which are highly sensitive to the kinematics of electron scattering, to uncover a giant intrinsic photocurrent response in pristine graphene. This photocurrent emerges exclusively at the charge neutrality point and vanishes abruptly at non-zero charge densities. Moreover, it is observed at places with broken reflection symmetry, and it is selectively enhanced at free graphene edges with sharp bends. Our findings reveal that the photocurrent relaxation is strongly suppressed by a drastic change of fast photocarrier kinematics in graphene when its Fermi level matches the Dirac point. The emergence of robust photocurrents in neutral Dirac materials promises new energy-harvesting functionalities and highlights intriguing electron dynamics in the optoelectronic response of Dirac fluids.