Just got to know that the Contao CMS system cannot use its Live-Update function, when the suhosin php hardening patch is active on the web server.
The Live-Update needs to unpack and execute so called phar files, which are by default disabled in suhosin. To be able to treat these phar files, an exception must be made in the php.ini configuration file:
; suhosin configuration
With this entry and a PHP reload, the Live-Update of Contao should now work.