Context. The evidence of line-like spectral features above 100 GeV, in particular at 130 GeV, which have been recently reported from some parts of the Galactic plane, poses serious challenges for any interpretation of this surprise discovery. It is generally believed that the unusually narrow profile of the spectral line cannot be explained by conventional processes in astrophysical objects, and, if real, is likely to be associated with dark matter. Aims. In this paper we argue that cold ultrarelativistic pulsar winds can be alternative sources of very narrow gamma-ray lines. Methods. We demonstrate that Comptonization of a cold ultrarelativistic electron-positron pulsar wind in the deep Klein-Nishina regime can readily provide very narrow (ΔE/E ≤ 0.2) distinct gamma-ray line features. To verify this prediction, we produced photon-count maps based on the Fermi LAT data in the energy interval 100 to 140 GeV. Results. We confirm earlier reports of the presence of marginal gamma-ray line-like signals from three regions of the Galactic plane. Although the maps show some structure inside these regions, unfortunately the limited photon statistics do not allow any firm conclusion in this regard. Conclusions. The confirmation of 130 GeV line emission by low-energy threshold atmospheric Cherenkov telescope systems, in particular by the new 28 m diameter dish of the H.E.S.S. array, would be crucial for resolving the spatial structure of the reported hotspots, and thus for distinguishing between the dark matter and pulsar origins of the "Fermi Lines".