Non-Hermitian skin effect (NHSE) is a unique feature studied extensively in non-interacting non-Hermitian systems. In this work, we extend the NHSE originally discovered in non-interacting systems to interacting many-body systems by investigating an exactly solvable non-Hermitian model, i.e., the prototypical Lieb-Liniger Bose gas with imaginary vector potential. We show that this non-Hermitian many-body model can also be exactly solved through generalized Bethe ansatz. By solving the Bethe ansatz equations accurately, the explicit eigenfunction is obtained, and the model's density profiles and momentum distributions are calculated to characterize the NHSE quantitatively. We find that the NHSE is gradually suppressed on the repulsive side but does not vanish as the repulsive interaction strength increases. On the attractive side, the NSHE for bound-state solutions is enhanced as interaction strength grows. In contrast, for scattering state the NHSE shows a nonmonotonic behavior in the whole interaction range. Our work provides the first example of the NHSE in exactly solvable many-body systems, and we envision that it can be extended to other non-Hermitian many-body systems, especially in integrable models.