Abundance of the cosmogenic nuclide chlorine-36 (36Cl) was measured together with the chloride (Cl−) concentration in different horizons of Quaternary permafrost samples collected from various types of ground ice in the northeastern part of Siberia. The 36Cl/Cl in 32 samples ranged in value from 2.4 × 10−14 to 1.4 × 10−12. Nonetheless, after a few extreme values were excluded, these 36Cl/Cl ratios provided a local permafrost chronometry. The general concordance of the modeled ages with geological expectations and other chronological methods supports the potential power of the proposed dating method. However, the large observed change in ratios from higher to lower values during the transition from Last Glacial Maximum to Holocene climatic conditions remains unexplained. An attempt to make use of the corresponding beryllium-10 (10Be) absolute concentrations in the same samples failed because input of 10Be attached to particulate matter into permafrost is unknown. Further 36Cl/Cl serial measurements of modern precipitation and fossil ground ice are needed to refine this dating method into a practical tool with a clear protocol.