Simulations of condensed matter systems at the hybrid density functional theory (DFT) level pose significant computational challenges. The elevated costs arise from the non-local nature of the Hartree-Fock exchange (HFX) in conjunction with the necessity to approach the thermodynamic limit (TDL). In this work, we address these issues with the development of a new efficient method for the calculation of HFX in periodic systems, employing k-point sampling. We rely on a local atom-specific resolution-of-the-identity scheme, the use of atom-centered Gaussian type orbitals (GTOs), and the truncation of the Coulomb interaction to limit computational complexity. Our real-space approach exhibits a scaling that is at worst linear with the number of k-points. Issues related to basis set diffuseness are effectively addressed through the auxiliary density matrix method (ADMM). We report the implementation in the CP2K software package, as well as accuracy and performance benchmarks. The method demonstrates excellent agreement with equivalent Gamma-point supercell calculations in terms of relative energies and nuclear gradients. Good strong and weak scaling performances, as well as GPU acceleration, make this implementation a promising candidate for high-performance computing.