The problem only appears when the facebook plugin is active. Obviously I disabled it in order to manage the site properly!
As I said before, I have deactivated several plugins and reactivated them all one by one. The Facebook for Woocommerce plugin is the only one having the effect described above. I’ve cleared my cached data on my site, as well as my browser. When I deactivate the Facebook plugin, everything works as expected.
The problem only happens when editing products. There is no fatal error in WooCommerce status.