The massive production of polymer-based respiratory masks during the COVID-19 pandemic has rekindled the issue of environmental pollution from nonrecyclable plastic waste. To mitigate this problem, conventional filters should be redesigned with improved filtration performance over the entire operational life while also being naturally degradable at the end. Herein, we developed a functional and biodegradable polymeric filter membrane consisting of a polybutylene adipate terephthalate (PBAT) matrix blended with cetyltrimethyl­ammonium bromide (CTAB) and montmorillonite (MMT) clay, whose surface properties have been modified through cation exchange reactions for good miscibility with PBAT in an organic solvent. Particularly, the spontaneous evolution of a partial core–shell structure (i.e., PBAT core encased by CTAB-MMT shell) during the electrospinning process amplified the triboelectric effect as well as the antibacterial/antiviral activity that was not observed in naive PBAT. Unlike the conventional face mask filter that relies on the electrostatic adsorption mechanism, which deteriorates over time and/or due to external environmental factors, the PBAT@CTAB-MMT nanofiber membrane (NFM)-based filter continuously retains electrostatic charges on the surface due to the triboelectric effect of CTAB-MMT. As a result, the PBAT@CTAB-MMT NFM-based filter showed high filtration efficiencies (98.3%, PM0.3) even at a low differential pressure of 40 Pa or less over its lifetime. Altogether, we not only propose an effective and practical solution to improve the performance of filter membranes while minimizing their environmental footprint but also provide valuable insight into the synergetic functionalities of organic–inorganic hybrid materials for applications beyond filter membranes.