Yes. A transitive verb takes an object. Here, the verb “ran” is only modified by the prepositional phrase “across the field.” The transitiveness or intransitiveness of a verb does not reside in the verb, but in how it is used in a clause. For example, if the word “ran” is used in this way: “He ran the company” it is a transitive verb because it takes a direct object, “company.” The way to determine whether a verb in a clause is transitive or intransitive is to reverse the syntax (word order). “The company was run by him”, but not “the field was run by the horse.” Prepositional phrases that modify a verb are sometimes confused with the “verb plus particle” construction: “He ran up the stairs” is a clause with the verb “ran” modified by the prepositional phrase “up the stairs”; but “ He ran up the bill” has a verb-plus-particle construction: “ran up” is the verb.