Before we get lost in the confusion of this discussion, something Rebbe Natan went to great lengths to clarify, but something I won't clarify here, he points out that still, this is only from the point of view of us created beings. From the standpoint of His Essence itself, there is no name, and nothing which exists beyond Himself.
Yep, it's really confusing but in a way I think we can use this teaching to explain how the Baal HaTanya (and I think the B"esht as well) could say the soul is a piece of the Godliness without being contradictory.
Yes, on the one hand, HaShem is devoid of 'pieces' but that is from the standpoint of His Essence. On the other hand, when it comes to His Name, which we know is the Holy Torah, so, it has letters, and we already know the souls of the Jewish People are the letters of the Holy Torah. So, it isn't hard to understand that saying that the Jewish soul is literally a part of Godliness, means: Each Jewish soul is rooted in a letter of the Holy Torah, HaShem's Holy Name, which exists in complete undivided unity with HaShem Himself.
I pray I've created a little less confusion and a little more understanding, at least in some of us, perhaps even myself. [As problematic as talking about something like this may be.]