In wireless localization services, a wideband spectrum is required for high resolution ranging. For this purpose, unlicensed bands have received substantial interest for their potential to reduce deployment cost. However, in the unlicensed spectrum, narrowband interference is often present and distorts band-limited reference signals for channel impulse response (CIR) estimation that is a key component to determine the location of users. In this paper, we propose a new Bayesian compressive sensing (BCS) framework to estimate complex-valued targets and apply it to mitigate the impact of subband interference on CIR estimation accuracy. Our Bayesian approach estimates the CIR by maximizing the posterior probability of the CIR from frequency domain signals in which a portion of the signal is corrupted by the interference. Based on the BCS framework, we propose three interference mitigation techniques that utilize the information on interfered subbands differently. We demonstrate the superior performance of the proposed schemes by showing improved ranging error statistics using measured indoor channels in the 5.8 GHz band.