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NBA star Iman Shumpert was INCREDIBLY calm for a guy who's fiancee unexpectedly gave birth in their home -- and you can hear the whole dramatic thing unfold on the 911 recording.

As we previously reported, the Cleveland Cavs star called 911 right after Teyana Taylor suddenly went into labor and gave birth to their daughter earlier this week.

You can hear Iman and Teyana are very emotional during the delivery -- and with help from the 911 operator, he was able to deliver a healthy baby girl.

After things calmed down, the 911 operator tells Iman ... "It's not everyday somebody delivers a baby on the phone."

True that.