Scattering problems in an isotropic homogeneous medium, such as the propagation of a pressure wave in acoustics and linear elastic waves in elastodynamics, are modeled here with an isogeometric boundary element method. The boundary element method facilitates wave radiation simulations in infinite exterior domains by reducing the problem to the scatterer's surface. This motivates an exact parametrization of the possibly curved manifold by connected spline patches. The unknown Cauchy data on this manifold are the solutions to boundary integral equations based on time-dependent convolution integrals. Using the convolution quadrature method (CQM), an approximate solution in the time domain is held by a quadrature rule with quadrature weights based on the solution to discretized elliptic problems in the Laplace domain \cite{c1}. Continuous and discontinuous higher-order NURBS basis functions span the approximative solution space to those discretized elliptic problems. Combining a multi-stage Runge-Kutta-based CQM with the spline approximations in the spatial variable enables an over all higher-order method in space and time. We measure the approximation quality by investigating the convergence rate in a combined space and time error norm based on uniform refinement in both variables. Applied is the direct Galerkin BEM with mixed boundary conditions \cite{c2}. Numerical experiments demonstrate the higher-order capability of the overall method. Also, implementational aspects are given, explaining the incorporation of IGA into existing integration routines using order elevation \cite{c3} and Bézier extraction algorithms \cite{c4}.