Knowledge of the phonon mean free path (MFP) holds the key to understanding the thermal properties of materials and nanostructures. Although several experiments measured the phonon MFP in bulk silicon, MFP spectra in thin membranes have not been directly measured experimentally yet. In this work, we experimentally probe the phonon MFP spectra in suspended silicon membranes. First, we measure the thermal conductivity of membranes with arrays of slits at different temperatures. Next, we develop a fully analytical procedure to extract the accumulated thermal conductivity as a function of the MFP. The measured phonon MFP in 145-nm-thick membranes with the surface roughness of 0.2 nm is shorter than that in bulk due to the scattering at the membrane boundaries. At room temperature, the phonon MFP does not exceed 400 nm. However, at 4 K, the MFP becomes longer, and some phonons can travel ballistically for up to one micrometer. These results thus shed light on the long-lasting question of the range of ballistic phonon transport at different temperatures in nanostructures based on silicon membranes.