This paper presents the design and analysis of gradient extremum seeking (ES) for scalar static maps, which are optimized in the presence of infinite-dimensional dynamics governed by Partial Diferential Equations (PDEs) of wave type containing a small amount of Kelvin-Voigt damping. This class of PDEs for extremum seeking has not been studied yet in the literature. We compensate the average-based actuation dynamics through a boundary controller via backstepping transformation. The local exponential convergence to a small neighborhood of the unknown optimal point is proven by means of an Input-to-State Stability (ISS) analysis as well as employing the averaging theory in infinite dimensions.