Is a woman obligated to inform her husband about money she possesses in the bank, and is it permissible for her to lie if asked about it, especially with her family advising secrecy for fear of the husband exploiting or demanding the money, and what is the legitimate Islamic ruling in this situation?

A woman has independent financial rights, and her husband has no authority over her money. She is not obligated to inform him about what she owns, and if she conceals her wealth from him, there is no sin upon her. Her concealing her money is not a betrayal. If her husband asks her, she has the option not to tell him, or to use tawriyah (deliberately using an ambiguous phrase that can be interpreted in two ways, one of which is true and the other is what the listener is likely to understand). Tawriyah is preferable to outright lying.