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The anisotropic optical response of the layered, nodal-line semimetal ZrSiS at ambient and high pressure is investigated by frequency-dependent reflectivity measurements for the polarization along and perpendicular to the layers. The highly anisotropic optical conductivity is in very good agreement with results from density functional theory calculations and confirms the anisotropic character of ZrSiS. Whereas the in-plane optical conductivity shows only modest pressure-induced changes, we found strong effects on the out-of-plane optical conductivity spectrum of ZrSiS, with the appearance of two prominent excitations. These pronounced pressure-induced effects can neither be attributed to a structural phase transition according to our single-crystal x-ray diffraction measurements, nor can they be explained by electronic correlation and electron-hole pairing effects, as revealed by theoretical calculations. Our findings are discussed in the context of the recently proposed excitonic insulator phase in ZrSiS.