Ultrafast acoustic imaging experiments are a powerful tool to investigate, at the nanometer scale, cell mechanical properties such as stiffness, viscosity, and adhesion, properties that play some roles in the life and death of cells. However, due to cell complex structures, the ultrafast acoustic signal analysis is quite intricate and depends on multiple parameters. Complex data analysis with poorly known parameters can be handled by a data clustering method as already shown in particle physics and biology. In this work, ultrafast acoustic data analysis is tackled by a spectral clustering method followed by a hierarchical agglomerating method. Coupled to conventional microscopy performed on the very same cell, the clustered data can be assigned to inner-cell features such as the nucleus, the cytoplasm, and the cytoskeleton. The signal dependency on the cell thickness and stiffness is highlighted. Moreover, thanks to the improvement of the signal-to-noise ratio, the nature of the adhesion is also assessed through observation and characterization of a polymerlike layer as thin as a few nanometers.