Xiao Guo (郭潇) earned the bachelor degree in Physics from Wuhan University in 2017 and PhD degree from National Astronomical Observatories, Chinese Academic of Science in 2023. Now he is a postdoctoral fellow in School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences. And he won the fellowship of China National Postdoctoral Innovation Talents Support Program in 2023. He mainly studies gravitational wave astrophysics including gravitational lensing of gravitational waves, detecting gravitational waves via pulsar timing array and so on. For more details, please see https://guoxiaowhu.github.io/Xiao_Guo/Dark matter (DM) occupies the majority of matter content in the universe and is probably cold (CDM). However, modifications to the standard CDM model may be required by the small-scale observations, and DM may be self-interacting (SIDM), warm (WDM) or fuzzy (FDM). Here we show that the diffractive lensing of gravitational waves (GWs) from binary black hole mergers by small halos (∼ 10^3 – 10^6 solar mass; mini-halos) may serve as a clean probe to the nature of DM, free from the contamination of baryonic processes in the DM studies based on dwarf/satellite galaxies. The expected lensed GW signals and event rates resulting from CDM, WDM, and SIDM models are significantly different from each other, because of the differences in halo density profiles and abundances predicted by these models. We estimate the detection rates of such lensed GW events for a number of current and future GW detectors. We find that Gravitational-wave Lunar Observatory for Cosmology (GLOC) may detect one such events per year assuming the CDM model, DECIGO (BBO) may detect more than several (hundreds of) such events per year, by assuming the CDM, WDM (with mass > 30 keV) or SIDM model, suggesting that the DM nature may be strongly constrained by DECIGO and BBO via the detection of diffractive lensed GW events by mini-halos. Other GW detectors are unlikely to detect a significant number of such events within a limited observational time period. However, if the inner slope of the mini-halo density profile is sufficiently steeper than the Navarro-Frenk-White (NFW) profile, e.g., the pseudo-Jaffe profile, one may be able to detect one to more than hundred such GW events by ET and CE. In addition, the GW rate lensed by FDM is too low to be detected in this mass range.