For more than 8 months, on Debian-based distributions (and others for sure), it's just impossible to run the software update to its latest version with the known dedicated command :
# update-flash-player-nonfree --install... which throws dozens of errors.
But today is special, with (again) a new version, I've decided to script the manual procedure described here.
Of course, the below script is supposed to run as root, so you may want to catch a glimpse of what it does before launching it
Check and get the sources here !
Feel free to write a comment on the GitHub's Gist page (or onto this very post) if another error occurred while you were dealing with it, I'll patch it A.S.A.P. .