We use electronic Raman scattering to study the low-energy excitations in BaFe$_2$(As$_{0.5}$P$_{0.5}$)$_2$ ($T_c approx 16$ K) samples. In addition to a superconducting pair breaking peak (2$Delta=6.7$ meV) in the A$_{1g}$ channel with a linear tail towards zero energy, suggesting a nodal gap structure, we detect spectral features associated to Pomeranchuk oscillations in the A$_{1g}$, B$_{1g}$ and B$_{2g}$ channels. We argue that the small Fermi energy of the system is an essential condition for these Pomeranchuk oscillations to be underdamped. The Pomeranchuk oscillations have the same frequencies in the B$_{1g}$ and B$_{2g}$ channels, which we explain by the mixing of these symmetries resulting from the removal of the $sigma_v$ and $sigma_v$ symmetry planes due to a large As/P disorder. Interestingly, we show that the temperature at which the peaks corresponding to the Pomeranchuk oscillations get underdamped is consistent with the non-Fermi liquid to Femi liquid crossover determined by transport, suggesting that the Pomeranchuk instability plays an important role in the low-energy physics of the Fe-based superconductors.
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