Natural biomaterials applicable for biomemristors have drawn prominent attention and are of benefit to sustainability, biodegradability, biocompatibility, and metabolism. In this work, multi-bit biomemristors based on the neutral polysaccharide dextran were built using the spin-casting method, which was also employed to explore the effect of dextran on the ternary biomemristic behaviors of dextran–chitosan nanocomposites. The doping of 50 wt% dextran onto the bio-nanocomposite optimized the ratio of biomemristance in high-, intermediate-, and low-resistance states (105:104:1). The interaction between dextran and chitosan (hydrogen-bond network) was verified by Fourier transform infrared (FTIR) and Raman spectroscopy analysis; through this interaction, protons derived from the self-dissociation of water may migrate under the electric field, and so proton conduction may be the reason for the ternary biomemristic behaviors. Observations from X-ray diffraction (XRD), thermogravimetric analysis (TGA), and differential scanning calorimetry (DSC) analysis displayed that the 50 wt% dextran/50 wt% chitosan nanocomposite had the greatest amorphous ratio as well as the highest decomposition and peak transition temperatures in comparison with the other three dextran–chitosan nanocomposites. This work lays the foundation for neutral biomaterials applied to green ultra-high-density data-storage systems.