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To support emerging applications ranging from holographic communications to extended reality, next-generation mobile wireless communication systems require ultra-fast and energy-efficient (UFEE) baseband processors. Traditional complementary metal-oxide-semiconductor (CMOS)-based baseband processors face two challenges in transistor scaling and the von Neumann bottleneck. To address these challenges, in-memory computing-based baseband processors using resistive random-access memory (RRAM) present an attractive solution. In this paper, we propose and demonstrate RRAM-based in-memory baseband processing for the widely adopted multiple-input-multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) air interface. Its key feature is to execute the key operations, including discrete Fourier transform (DFT) and MIMO detection using linear minimum mean square error (L-MMSE) and zero forcing (ZF), in one-step. In addition, RRAM-based channel estimation as well as mapper/demapper modules are proposed. By prototyping and simulations, we demonstrate that the RRAM-based full-fledged communication system can significantly outperform its CMOS-based counterpart in terms of speed and energy efficiency by $10^3$ and $10^6$ times, respectively. The results pave a potential pathway for RRAM-based in-memory computing to be implemented in the era of the sixth generation (6G) mobile communications.