Even if possible, there are still a lot of other problems to get through before this technology can be useful in the mainstream.  For one thing, the cost and development time is very prohibitive currently.  I suppose that’s true of most space technologies!

According to scientists at prestigious Cambridge University, yes, because versatile but brittle carbon nanotubes can now be softened and elongated. The softening technology seems to still be top-secret since I can't track down a press release or study publication, but the elongation is accomplished by joining multiple single nanotubes together. NASA has requested 144,000 miles of carbon nanotube tether to provide the pull for an elevator car to be moved along a track to a space station. NASA offered a $4 million prize for this breakthrough because they must retire their shuttle fleet and don't have the funding (which obviously will exceed $4 million) to replace the retired fleet with a new generation of shuttles.

"This isn't going to happen for probably the next decade at least, but in theory this is now possible ... [because of] advance in [nanotube] materials for the tether." (John Winter of EuroSpaceWard. Phys.org)