Collisionless shock acceleration of carbon ions (C^{6+}) is investigated in the ultrarelativistic regime of laser-plasma interaction by accounting for the radiation reaction force and the pair production in particle-in-cell simulations. Both radiation reaction force and pair-plasma formation tend to slow down the shock velocity, reducing the energy of the accelerated ions, albeit extending the timescales of the acceleration process. The slab plasma target achieves a lower energy spread while the target with a tailored density profile yields higher ion acceleration energies.