You can prove your skills as programmer/developer without writing automated tests. Learn tools used for other testing-related tasks: load testing, etc. Become expert system administrator. Tools like Jenkins. Logstash to analyze logs (to be more productive as manual tester).
Prove your skills by writing scripts needed to manage the test data, like anonymization of data when copied from PROD to QA.
When you are competent programmer, test automation is just using yet another library.
Python would be especially good language to learn as system administrator, and also for test automation. And there are many good online courses, including certified ones. Then, try test automation in your own free time.
When you have the skills, you can either ask for a transfer to test automation, or find a job which is better match for your career interests in another company.
You can talk to your manager that you would like to learn new skills. Of course for this to work, you need to be in a good standing with your manager using your current skillset. Companies have good reason to allow current employees to grow skills (to make them happy and not leave), but it is your own responsibility to manage your own career. Being aware of what you want is a good first step.
If you are pigeon-holed in manual testing with no way to enhance your career according to your interests, you can look for another position of a manual tester where automation is not banned. This will be something to negotiate in new job offer.