Is a wife considered Islamically divorced if she tells her husband, "I am determined to get a divorce, and from this moment on, I consider myself divorced," and the husband replies, "As you wish"?

The husband's statement "as you wish" is an ambiguous phrase (kinayah) that could imply divorce or something else. If he intended to effect a divorce by it, then it occurs; otherwise, nothing happens with this phrase. Divorce does not occur through ambiguous phrasing without the intention, even if uttered in a state of anger or if the wife asks for divorce. The fact that the marriage is not officially documented has no bearing on this ruling, though it is important to emphasize the significance of documenting marriage in this era.