I have been there first time.
The reason why I think this might not be natural is that the expression "first time" seems to describe "past". You do not use the present perfect with an adverb indicating "past", in my understanding.
However, I feel the phrase below is not that funny.
This is the first time that I have been there.
I am wondering about the relationship between "the first time" and "I have been there". "That" could behave like a relative adverb. If that is the case, the antecedent of "that"—"the first time"—is shared with the following clause: "I have been there". Then, is it OK to say "I have been there first time"?