The correct option is B who lives across the street
An adjective clause is a group of words that contain a subject and a verb, and it functions as an adjective that modifies a noun or a pronoun. Adjective clauses usually begin with relative pronouns such as who, whom, whose or a relative adverb like when, where and so on.
Option B is correct as it is an adjective clause containing a subject 'who' (which is a relative pronoun), a verb 'lives' and modifies the noun "lady".
Option A is incorrect because it is a phrase, not a clause, it does not modify the noun.
Option C is incorrect because this is also a phrase.
Option D is incorrect because the answer lies in B.