In this paper, we explore the possibility of using free-standing thin films from single-walled carbon nanotube (SWCNT) material in optics of the extreme ultraviolet (EUV) range. Test samples were fabricated using an aerosol chemical vapor deposition method. Synchrotron radiation was used to record the transmittance spectra of samples in the EUV range. The measured transmittance for a film 40 nm thick almost monotonously increases from 76% at a wavelength of 20 nm–99% at a wavelength of 1 nm. The measured stress-strain curve for the test samples shows that the SWCNT-based thin films have rather high ductility as opposite to fragile films made of conventional solid state materials. We use numerical simulations to demonstrate that the film strain occurs mainly by straightening and sliding of the nanotubes past each other without forming of strain localization responsible for fragile behavior. The combination of high radiation transmittance and unique mechanical properties makes the SWCNT-based thin films very promising for use in the EUV optics. In particular, such films can be used to protect delicate optical elements for EUV lithography from their contamination with debris particles.