Is it permissible to remain employed at an interest-based bank due to the unavailability of an alternative, and because of family and financial obligations, even though the wife works and has her own income?

It is obligatory to leave work in a Riba-based bank immediately, unless there is a necessity that prevents it. If such a necessity exists, where basic needs cannot be met otherwise, then it is permissible to remain in this forbidden work until a permissible job becomes available. And whoever fears Allah and relies upon Him, He will provide for him from where he does not expect.