Since I am processing user's IP address when they make a connection, does GDPR require me to notify the user of such an action in, say a banner shown upon accessing the server?
Maybe. Can you identify specific users just from their IP address? It seems unlikely, especially if they are merely connecting to the server without being able to log in. In such an instance, it is arguable that you are not "processing" personal data so there is no requirement to notify, etc.
Bear in mind that a lot of automated bots will be port scanning for open SSH servers and they cannot be regarded as "natural persons" for the purpose of the legislation. The number of such bots would seem likely to outweigh the number of natural persons accessing the server in most cases, so it would be sufficient to only show the banner after logging in.
am I obliged to treat possible login attemps and the data entered as personal data?
It depends. If most of the attempts are automated, they are not connected to a natural person, but the actual data entered might count as personal data if users are issued specific credentials. Otherwise, if they log in with a generic account name and password, arguably it is not personal data since it does not specifically identify a person.
What if the interface processes user data, but its technology does not necesarily provide for a human-readable interaction, such as an ICMP response?
I don't know enough about how this works to provide an answer, I'm afraid. I imagine it could be caught by the same provisions, but it would be a lot more fact-specific.