The low-frequency noise in a nanometer-sized virtual memristor consisting of a contact of a conductive atomic force microscope (CAFM) probe to an yttria stabilized zirconia (YSZ) thin film deposited on a conductive substrate is investigated. YSZ is a promising material for the memristor application since it is featured by high oxygen ion mobility, and the oxygen vacancy concentration in YSZ can be controlled by varying the molar fraction of the stabilizing yttrium oxide. Due to the low diameter of the CAFM probe contact to the YSZ film (∼10 nm), we are able to measure the electric current flowing through an individual filament both in the low resistive state (LRS) and in the high resistive state (HRS) of the memristor. Probability density functions (Pdfs) and spectra of the CAFM probe current in both LRS and HRS are measured. The noise in the HRS is found to be featured by nearly the same Pdf and spectrum as the inner noise of the experimental setup. In the LRS, a flicker noise 1/fγ with γ ≈ 1.3 is observed in the low-frequency band (up to 8 kHz), which is attributed to the motion (drift/diffusion) of oxygen ions via oxygen vacancies in the filament. Activation energies of oxygen ion motion determined from the flicker noise spectra are distributed in the range of [0.52; 0.68] eV at 300 K. Knowing these values is of key importance for understanding the mechanisms of the resistive switching in YSZ based memristors as well as for the numerical simulations of memristor devices.