The new class of fully silicon-compatible hafnia-based ferroelectrics with high switchable polarization and good endurance and thickness scalability shows a strong promise for new generations of logic and memory devices. Among other factors, their competitiveness depends on the power efficiency that requires reliable low-voltage operation. Here, we show genuine ferroelectric switching in HfxZr(1-x)O2 (HZO) layers in the application-relevant capacitor geometry, for driving signals as low as 800 mV and coercive voltage below 500 mV. Enhanced piezoresponse force microscopy with sub-picometer sensitivity allowed for probing individual polarization domains under the top electrode and performing polarization domains under the top electrode and performing a detailed analysis of hysteretic switching. The authentic local piezoelectric loops and domain wall movement under bias attest to the true ferroelectric nature of the detected nanodomains. The systematic analysis of local piezoresponse loop arrays reveals a totally unexpected thickness dependence of the coercive fields in HZO capacitors. The thickness decrease from 10 to 7 nm is associated with a remarkably strong decrease of the coercive field, with about 50% of the capacitor area switched at coercive voltages <= 0.5 V. Our explanation consistent with the experimental data involves a change of mechanism of nuclei-assisted switching when the thickness decreases below 10 nm. The practical implication of this effect is a robust ferroelectric switching under the millivolt-range driving signal, which is not expected for the standard coercive voltage scaling law. These results demonstrate a strong potential for further aggressive thickness reduction of HZO layers for low-power electronics.