We consider the motion of rigid bodies in a potential fluid subject to certain nonholonomic constraints and show that it is described by Euler-Poincare-Suslov equations. In the two-dimensional case, when the constraint is realized by a blade attached to the body, the system provides a hydrodynamic generalization of the classical Chaplygin sleigh problem, one of the best known examples of nonholonomic systems. The dynamics of the generalized sleigh is studied in detail. Namely, the equations of motion are integrated explicitly, and the asymptotic behavior of the system is described analytically and from the qualitative point of view. It is shown that the presence of the fluid brings new features to such a behavior.