We explore the possibility that CPT symmetry is broken in the neutrino sector at a high enough energy, using for this the high-energy neutrino flux that is expected to come from distant astrophysical sources such as active galaxies and gamma-ray bursts. CPT violation (CPTV) can affect neutrino oscillations either by modifying the oscillation length or its amplitude. We show that when only the oscillation length is affected, such as when CPTV is introduced through a modified dispersion relation, the effects on the astrophysical neutrino flavour fluxes are not visible. Hence, in order to modify also the oscillation amplitude, CPT is instead violated by adding a nondiagonal energy-independent contribution to the standard, mass-driven, neutrino oscillation Hamiltonian, within the context of the Standard Model Extension, and allowing it to become comparable in magnitude to the latter at a scale of 1 PeV. The CPTV term introduces three new mixing angles, two new eigenvalues and three new phases, all of which have currently unknown values. We have allowed these new parameters to vary, together with the ones associated to pure standard oscillations, and explored the consequences on the flavour fluxes. Detection prospects at the IceCube neutrino telescope, and in a fictional larger detector, have been studied, assuming different models for the neutrino flux. Our results suggest that, when the flavour fluxes at production are in the ratios φ0e : φ0μ : φ0τ = 1 : 2 : 0 or 0 : 1 : 0, the modifications to the flavour fluxes at Earth are larger and more clearly separable from the standard-oscillations predictions. When the ratios are 1 : 0 : 0, the separation is less clear. We conclude that, for certain values of the CPTV parameters, IceCube might be able to detect potentially large deviations from the standard flavour fluxes, even for flux models as low as the Waxman-Bahcall bound. Precision measurements of these parameters, however, are unlikely to be made unless a much larger Cerenkov detector or a one of a different kind are used.