"Barking" is a single gerund, the subject of the verb "woke." It is not a phrase.
INSISTENT BARKING
No, "barking" in the sentence you provided is not a gerund. A gerund is a verb form that functions as a noun and ends in "-ing." In this case, "barking" is part of the verb phrase "woke me up," where "barking" is the present participle form of the verb "bark," serving as a modifier to describe the type of action that the dogs were engaged in. It describes the action of the dogs while they were barking, rather than functioning as a noun.
