We study the behavior of an isolated field-oriented chain of dipolar particles in elongational fluid flow. Our main goal is to emphasize the effect of dipolar interactions on the chain's contribution to the pressure tensor and to the viscosities of a dilute suspension of these linear aggregates. In our model, despite the overall rigid appearance of the chain at rest, the constituent beads may move slightly relative to one another, conferring a certain degree of flexibility to the chain. This flexibility is quantified in terms of a dimensionless parameter, lambda(-1), comparing thermal and dipolar energies. We perform an expansion in lambda(-1), and obtain the first correction to the rigid chain contribution to the Kramers' pressure tensor for different flow geometries. The interplay of the elongational flow field and the field-induced chain orientation gives rise to a rich variety of scenarios. We compute the elongational, shear, and rotational viscosities in some representative situations.