The sentence in which the underlined verb is in the future perfect progressive tense is the last option.
-She will have been visiting New York City for a week when she turns five years old.
The future perfect progressive is also refered to as the future perfect continuous, This tense describes actions that will continue up until a point in the future. The future perfect continuous patter is will + have + been + the verb’s present participle (verb root + -ing).