What is the ruling of Islamic law regarding a father's refusal to marry off his daughter to a righteous man whom they both are pleased with and wish to marry for the sake of chastity, merely because of an age difference between them? And is it permissible for the daughter in this case to appoint someone other than her father to be her guardian?

A guardian does not have the right to prevent his female ward from marrying without a legitimate religious justification if a suitable suitor proposes to her. Otherwise, he would be unjustly preventing her marriage. A significant age difference is not a valid reason for prevention. It is the woman's right to present her case to the Sharia court if her guardian prevents her marriage, so that the judge may marry her. It is not permissible for her to appoint anyone other than the judge. It is advisable to minimize differences between spouses to achieve harmony. If marriage to this woman is not feasible, then the relationship with her must be severed, as she is a non-mahram.