Suspensions of copper nanowires are emerging as new electronic inks for next-generation flexible electronics. Using a novel surface acoustic wave driven extensional flow technique we are able to perform currently lacking analysis of these suspensions and their complex buffer. We observe extensional viscosities from 3 mPa s (1 mPa s shear viscosity) to 37.2 Pa s via changes in the suspension concentration, thus capturing low viscosities that have been historically very challenging to measure. These changes equate to an increase in the relative extensional viscosity of nearly 12 200 times at a volume fraction of just 0.027. We also find that interactions between the wires and the necessary polymer additive affect the rheology strongly. Polymer-induced elasticity shows a reduction as the buffer relaxation time falls from 819 to 59 ms above a critical particle concentration. The results and technique presented here should aid in the future formulation of these promising nanowire suspensions and their efficient application as inks and coatings.