In-band Full Duplex (IBFD) wireless systems have attracted attention for use in emerging wireless systems as they provide significant enhancement to system capacity over traditional techniques. In IBFD, transmission and reception occur simultaneously and on the same frequency band. Thus, the main challenge in system design is self-interference (SI) where the strong transmit signal leaking to the receiver interferes with the weak received signal. SI typically includes a nonlinear component resulting from the transmitter Power Amplifier (PA); therefore, the resulting SI is highly dependent on transmit power. In this paper, an SI cancellation (SIC) scheme for IBFD systems is presented and verified. The proposed nonlinear SIC scheme is based on using a Radial Basis Function Neural Network (RBFNN) to model the nonlinear SI. A practically motivated feature representation which includes instantaneous power of the transmit signal is introduced, enabling efficient modelling of amplitude-dependent and memory-dependent nonlinearities. The proposed RBFNN canceller offers localized approximation capability and inherently stable training structure, making it suitable for addressing the SI dependence on transmit power. Simulation results demonstrate that the RBFNN-based SI canceller outperforms model-based cancellers such as the Memory Polynomial (MP) cancellers by 10 dB across a transmit power range of 30 dB.