In this paper, we investigate the fluid motion and shape of a non-Newtonian drop, modeled as a Chilcott-Rallison fluid, in the steady, uniaxial extensional flow of an unbounded Newtonian liquid. We consider two values of the capillary number, Ca = 0.05 and 0.10, below the critical value for continuous elongation of a Newtonian drop in the same flow. Solutions are then obtained for two values of the Chilcott-Rallison extensibility parameter, L-2 = 144 and 600, appropriate roughly for polymer solutions in the Boger fluid range of concentration, each for several values of the viscosity ratio lambda, and the polymer concentration, c. It is shown that the main region of impact of the polymer on the flow inside the drop occurs on the central symmetry axis, and near the tips of the drop. The effect of polymer on the drop shape is complicated. In general, there is a balance between the influence of the viscoelastic tensile stress contribution in the direction of the symmetry axis, which tends to pull the interface inward and thus decrease the flow-induced deformation at the ends of the drop, and modifications of the viscous stress due to changes in the flow and the pressure due viscoelasticity, both of which tend to increase deformation. For L-2 = 600, the tensile stress effect is dominant and the viscoelastic drop is less deformed overall and has lower curvature at the tip than for a Newtonian drop at the same capillary number. For LL = 144, however, this trend is reversed and the effect of viscoelasticity is to increase deformation and increase the curvature at the end of the drop.