As far as I can tell, exporting data has been removed from the user interface, and documentation on the topic is outdated.
The current procedure for those of us who do not fully trust Withings to be around forever, and therefor wish to periodically backup our data, seems to be to point our browsers at https://account.withings.com/export/user_select to request a download.
A similar to the link is to be found at https://support.withings.com/hc/en-us/articles/201491377-Health-Mate-Online-Dashboard-Exporting-my-data, but just as most of the text in that article it is obsolete and points at a Nokia Health domain. Oddly enough that link still will yield in a downloadable backup, yet with files lacking all data since mid-January 2018. Downloads from the withings.com domain seem to contain all data.
Hope this helps someone.
(Please do not mistake my username for being a Withings official. It was assigned to me from the email address I signed up with, and I believe it can not be changed.)