Energetic gamma rays scatter on soft background radiation when propagating through the Universe, producing electron-positron pairs (A.I. Nikishov, Sov. Phys. JETP 14, 393 (1962)). Gamma rays with energies between 100 GeV and a few TeV interact mostly with infrared background photons whose amount is poorly known experimentally but safely constrained from below by account of the contribution of observed light from known galaxies (R.C. Keenan, A.J. Barger, L.L. Cowie, and W.-H. Wang, Astrophys. J. 723, 40 (2010); arXiv: 1102.2428). The expected opacity of the intergalactic space limits the mean free path of TeV gamma rays to dozens of Megaparsecs. However, TeV photons from numerous more distant sources have been detected (S.P. Wakely and D. Horan, http://tevcat.uchicago.edu/ ). This might be interpreted, in each particular case, in terms of hardening of the emitted spectrum caused by presently unknown mechanisms at work in the sources (S. Archambault et al. (VERITAS and Fermi LAT Collaborations), Astrophys. J. 785, L16 (2014); arXiv: 1403.4308). Here we show that this interpretation is not supported by the analysis of the ensemble of all observed sources. In the frameworks of an infrared-background model with the lowest opacity (R.C. Gilmore, R.S. Somerville, J.R. Primack, and A. Dominguez, Mon. Not. Roy. Astron. Soc. 422, 3189 (2012); arXiv: 1104.0671), we reconstruct the emitted spectra of distant blazars and find that upward spectral breaks appear precisely at those energies where absorption effects are essential. Since these energies are very different for similar sources located at various distances, we conclude that the breaks are artefacts of the incorrect account of absorption and, therefore, the opacity of the Universe for gamma rays is overestimated even in the most conservative model. This implies that some novel physical or astrophysical phenomena should affect long-distance propagation of gamma rays. A scenario in which a part of energetic photons is converted to an inert new particle in the vicinity of the source and reconverts back close to the observer (M. Simet, D. Hooper, and P. Serpico, Phys. Rev. D 77, 063001 (2008); arXiv: 0712.2825; M. Fairbairn, T. Rashba, and S. Troitsky, Phys. Rev. D 84, 125019 (2011); arXiv:0901.4085) does not contradict our results. This new axion-like particle appears in several extensions of the Standard Model of particle physics (J. Jaeckel and A. Ringwald, Ann. Rev. Nucl. Part. Sci. 60, 405 (2010); arXiv: 1002.0329) and may constitute the dark matter (P. Arias et al., JCAP 1206, 013 (2012); arXiv: 1201.5902).