Modern binaural hearing aids utilize multimicrophone speech enhancement algorithms to enhance signals in terms of signal-to-noise ratio, but they may distort the interaural cues that allow the user to localize sources, in particular, suppressed interfering sources or background noise. In this paper, we present a novel algorithm that enhances the target signal while aiming to maintain the correct spatial rendering of both the target signal as well as the background noise. We use a bimodal approach, where a signal-to-noise ratio (SNR) estimator controls a binary decision mask, switching between the output signals of a binaural minimum variance distortionless response (MVDR) beamformer and scaled reference microphone signals. We show that the proposed selective binaural beamformer (SBB) can enhance the target signal while maintaining the overall spatial rendering of the acoustic scene.