In the framework of the design study of future linear colliders, the Compact Linear Collider (CLIC) aims for electron-positron collisions with high luminosity at a nominal center-of-mass energy of 3 TeV. To achieve the luminosity requirements, predamping rings (PDRs) and damping rings (DRs) are required: they reduce the beam emittance before the beam is accelerated in the main linac. Several kicker systems are needed to inject and extract the beam from the PDRs and DRs. In order to achieve both low beam coupling impedance and reasonable broadband impedance matching to the electrical circuit, striplines have been chosen for the kicker elements. In this paper, we present the complete design of the striplines for the DR extraction kicker, since it is the most challenging from the field homogeneity point of view. The excellent field homogeneity required, as well as a good transmission of the high voltage pulse through the electrodes, has been achieved by choosing a novel electrode shape. With this new geometry, it has been possible to benefit from all the advantages that the most common shapes introduce separately. Furthermore, a detailed study of the different operating modes of a stripline kicker allowed the beam coupling impedance to be reduced at low frequencies: this cannot be achieved by tapering the electrodes. The optimum design of the striplines and their components has been based on studies of impedance matching, field homogeneity, power transmission, beam coupling impedance, and manufacturing tolerances. Finally, new ideas for further improvement of the performance of future striplines are reported.