See also examples sentences of 先 in the dictionary I provided. The particle に (or any other particles) is attached to the preceding noun 先, not the following one 体, and 先に works as an adverbial phrase. 手より先に体の方(が) (=手より体の方(が)先に) is a comparison "which is earlier" there, thus, it means "body (does/did) earlier than hand(s) (do/did)".
Judging from your way of interpretation of the phrases in this thread, and "開け放た would make the verb negative" in your previous thread, you haven't learned Japanese grammar properly via textbooks or something so far, right? I strongly recommend getting a well-structured textbook and learning grammar with using it. There were members who learned Japanese their own way, and their Japanese didn't improve for many years as a result. It might not be impossible to learn Japanese via translating real materials as you would try to do, but it's no doubt an inefficient way of learning. My two cents.