This paper is an experimental investigation of the deformation and relaxation of a Newtonian drop suspended in a PIB/PB Boger fluid. The suspending fluid is undergoing a planar extensional flow produced in a four-roll mill. We show that increasing elasticity of the suspending fluid has a pronounced effect on both the deformation and relaxation of a drop. For steady flows, as the strength of viscoelastic effects in the suspending fluid is increased, the drops become more deformed, with ends that are generally more pointed. This leads to a decrease in the maximum ("critical") capillary number for the existence of a steady, deformed drop shape. In transient startup and step flows, the elasticity of the suspending fluid produces a large deformation shape that is more pointed at its ends and more tubular in its midsection than is observed for a drop in a Newtonian fluid (bulbous ends with necking at the waist). This enables a drop in the PIB/PB suspending fluid to be extended to a longer length without breaking upon flow cessation. However, at smaller deformations, the elasticity of the suspending fluid retards the relaxation of the drop. The observed viscoelastic effects on the steady and transient deformation, as well as the relaxation of drops in the PIB/PB suspending fluid, cannot be explained by viscoelastic modifications of the global, undisturbed flow field. Instead, our results suggest the existence of a non-linear coupling between the drop shape, the local disturbance flow, and the polymer configuration in the vicinity of the drop. This coupling enhances elastic effects, such that a drop can display significant non-Newtonian behavior prior to any changes in the global, undisturbed flow field.