Use .fi domains and you absolutely can. You really don't even need to code, run or host anything either unless you wanted to. Probably goes for some other lesser known ccTLDs too I'd assume.
Register as a private individual and leave the field where they ask if you want your contact info listed for consumers unchecked. Then, once they process the application and you're a registrar, never register domains for anyone but yourself.
The bit about not needing to host or code anything was a reference to the fact that Traficom maintains a registrar web interface[4]. If you're a low volume registrar, which you should be since you're a registrar for yourself, the web interface is all you need.
Now, couple of things I'm uncertain about: whether or not the registration process involves fees for non-citizens, and which payment methods they support besides SEPA transfers.
[0]: https://www.traficom.fi/en/communications/fi-domains
[1]: https://www.traficom.fi/en/communications/fi-domains/domain-...
[2]: https://www.traficom.fi/sites/default/files/media/file/regis...
[3]: https://www.traficom.fi/en/services/registration-registrar