What is the נשמה, our soul?
Of course, our knowledge to precisely define and delineate certain things is limited, but we do have a מסורה (tradition) from our חכמים, from way back, דורי דורות, which finds expression in Chazal and ancient tefillos. And that is, that it is a spiritual creation of הקב"ה, that comes from a very high place (sometimes a description of חצובה מתחת כסא הכבוד, hewn from under the heavenly throne of glory, is seen). In later generations, some have used an expression taken from a pasuk in ספר איוב (לא:ב) for it, namely חלק אלו-ה ממעל. Chabad-Lubavitch adds the word ממש (literally) to that, zealously stressing it in its preaching. Due to that, there is fairly widespread misconception, to a significant degree fueled by and emanating from them, that the neshamah is "a piece of G-d" literally, ר"ל.
Of course, that is quite problematic, as it blurs, or eliminates, the distinction between the Creator and the created, the בורא and the נברא, and is reminiscent of foreign faiths, עבודה זרה ר"ל.
The matter has been discussed online in various fora, at various times, such as here, here, and here.
Recently, during a shiur on the ימים נוראים davening, in speaking about עלינו, a speaker displayed a Chabad-Lubavitch siddur open to that prayer (סדור תהלת ה' עם תרגום באנגלית, תשע"ד, ניו יורק, עמוד 288), in which the translation states that Hashem has given each of us a neshamah "that is part of Himself". Such dangerous, misleading, if not outright heretical words, blur or eliminate the distinction between the בורא, הקב"ה (the creator, Almighty G-d), and the נברא (the created - man), a very dangerous thing, which is even more problematic in a tefillah where we stress our distance from ע"ז, especially during this awesome time of the year.
On the other hand, in the beginning of the tefillah of אמיץ כח, in mussaf of Yom Kippur, where it relates the early history of the world, it states with regard to the neshamah (line 12), הפחת בחלדו טהר נשם מזבולך, You (HKB"H) breathed into him a pure breath (neshamah) from your zevul (an area of the heavenly sphere). No mention of it being a "part".
In the zechus of being careful, understanding what we say and believe, particularly in the area of עיקרי הדת, and avoiding and resisting dangerous distortions of our faith and mesorah, in addition to outright אפיקורסות (even if spread by men with lengthy beards), may we be zoche to a שנה טובה ומבורכת.