  We demonstrate that a proton irradiation with fluences of
$3.6\times10^{10}$/cm$^{2}$ at low energy ($<$ 36 MeV) and $1.46 \times
10^{10}$/cm$^{2}$ at high energy (40 MeV and 90 MeV combined) on the dielectric
mirrors of Fabry-P\'erot cavities with a finesse of about 700 000 causes less
than 5% change in the finesse. Furthermore, no influence on the coupling
efficiency to the cavities was observed, the efficiency being approximately
70%. The irradiation was carried out with a spectrum approximating the proton
energy spectrum of a highly elliptic Earth orbit with duration of 5 years,
proposed for the Space-Time Explorer and Quantum Equivalence Space Test
(STE-QUEST) mission [\url{http://sci.esa.int/ste-quest/}].
