An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline weighted essentially non-oscillatory (WENO) reconstruction with high order flux evaluation. Polyharmonic splines are utilized in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible, and optimal in the associated Sobolev (Beppo-Levi) space. The flux evaluation is accomplished by solving generalized Riemann problems across cell interfaces. The mesh adaptation is performed through an a posteriori error indicator, which relies on the polyharmonic spline reconstruction scheme. The performance of the proposed method is illustrated by a series of numerical experiments, including linear advection, Burgers's equation, Smolarkiewicz's deformational flow test, and the five-spot problem.