It may be possible to send a speculative or unsolicited application even if there is not a current job advertisement. Sometimes this will be stated on the organization’s website as well. For an unsolicited internship application, you can find some more information and examples for what must necessarily be included here, if the organization does not otherwise specify:

• https://en.life-in-germany.de/unsolicited-application-in-germany/
• https://jobs.makesense.org/en/media/guide-candidat/comment-realiser-une-candidature-spontanee/

Some general advice on sending unsolicited internship applications is also available on the website of the University of Potsdam.

It is not necessarily advisable to send unsolicited applications to government departments or ministries if not otherwise advertised on their career websites, however, if you have time and energy to spare and this is truly your dream internship, nothing should stop you from reaching out.