Short answer: Yes, but no. Medium answer: In real life, we say “I’m the only friend you’ve got.” “It is I who am” is technically correct, since “who” refers back to the previously mentioned person, for example, “it is they who are (not “is”) coming”. This would be obviously in a more heavily inflected Western European language such as German or Spanish. In real life, native English speakers simply avoid that needlessly complicated construction, and don’t really know what the supposedly correct verb form is. Non-natives, on the other hand, would…. Conclusion: Forget about “It is I who” — nobody talks or writes that way in the twenty-first century.

I am the only friend you've got / I'm the only friend you've got

I think the best way to write this sentence, is to write it like: "I'm the only friend you've got." and maybe even better: "I'm the only friend you have." but if you really want to write it like that, you should write it like: "It is I, who is the only friend you've got." I hope this comment helped! :)

No, it's correct, because "I am". Am is the first person singular of the verb of being. The sentence might sound a bit more natural if you said "I'm the only friend you've got."

I'm the only friend you got/ it is me who is the only friend you got but the first one makes more sense.