Seems like a simple question, but a lot of people ask it. The simple answer is no. Without lye, there is no soap. Soap, by definition, is the result of a chemical reaction between fatty acids from animal or vegetable oils and lye called saponification. Glycerin is also a by-product of this reaction. Thus, most handmade soap is about 75% soap and 25% glycerin. There is no lye remaining in a properly made soap.
So why do you see products that act like soap without any lye in them? These are detergents. They are made with surfactants which lower the surface tension of water, essentially making it ‘wetter’ so that it is less likely to stick to itself and more likely to interact with oil and grease. Most shower gels, and even so-called “soaps” you buy at the store are made with surfactants that dry out your skin.
If you are looking to make soap without lye as a fun project, you might consider using a melt-and-pour soap base from your local craft supply store. They are usually a combination of soap and surfactants. If, however, your goal is to make soap without lye that is more moisturizing than store-bought soap, I would look for a quality melt-and-pour base from a reputable soapmaking supply store.
Or you could buy some of my handmade soap made from scratch with fresh goat’s milk and skin-friendly vegetable oils and butters (and of course, lye).