She wants convincing proof whether she is obliged to return a book she borrowed, even if that is ten years later

The Islamic ruling for anyone who borrows something, whether that is a book – as in your question – or anything else, and whether one year has passed or seventy years, is that it is obligatory to render back the trust to those to whom it is due, and to repent from falling short in returning the borrowed item to its owner. We think that rulings such as this are quite clear; in fact this is a given among all people, for several reasons, each of which is sufficient to convince you if you think about it. These reasons include the following: 

Firstly: 

The teachings of Islam enjoin that. We have previously quoted a great deal of evidence in fatwa no. 83599. We would add here further clear evidence: it was narrated from al-Hasan, from Samurah ibn Jundub (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “The borrower owes what he took until he returns it.” Narrated by Imam Ahmad in al-Musnad (33/277). The commentators on the Mu’sasat ar-Risaalah edition said: It is hasan because of corroborating evidence. Adh-Dhahabi said: Its isnaad is good. Al-Muhadhdhab (7/3415). Ibn al-Mulaqqin said: [It is saheeh] according to the conditions of al-Bukhaari. Al-Badr al-Muneer (6/753)

In ‘Awn al-Ma‘bood (9/344) it says: 

This hadeeth indicates that the individual is obliged to return that which he took of the property of others, whether that was by borrowing or renting or otherwise. (This obligation remains in effect) until he returns it to its owner. End quote. 

Secondly: 

The moral and ethical values on which people, believers and disbelievers, young and old, rich and poor, knowledgeable and ignorant, are agreed dictate fulfilling trusts and returning all things to their owners, whether the property in question is small or great. The one who is careless and devours a little unlawfully will go on to transgress with regard to greater amounts, and he will transgress the limits set by Islam, and will pay no heed to ethics and values. 

Thirdly: 

Imagine if everyone who borrowed something never returned it at all. How would the world be in that case? How would people interact with one another when no one could trust anyone else? Undoubtedly is an alarming scenario, if it were to happen – Allah forbid. 

Conversely, you can imagine a society in which all people render trusts back to those to whom they are due, whether the matter was small or great. What a secure and peaceful life people would live in that case. How many other virtues would this virtue lead to! Undoubtedly that would be an ideal world which all of us would hope for. The Prophet (blessings and peace of Allah be upon him) has told us of some of the virtues of the righteous among the earlier generations, and he told us that one of them said (when they were trapped in a cave after a rock fell and blocked the mouth of the cave, and he and his companions sought help from Allah by virtue of righteous deeds that they had done): “O Allah, I hired a man in return for a measure (faraq) of corn. I offered it to him but he refused to take it. So I sowed that corn and cultivated it, until I had acquired cattle and a herdsman thereby (by means of that investment). Then he came to me and said: O slave of Allah, give me my dues. I said: Go to those cattle and their herdsman, for they are yours. He said: Are you making fun of me? I said: I am not making fun of you; they are yours. O Allah, if You know that I did that seeking thereby Your Countenance, then grant us relief. And they were granted relief.” 

Narrated by al-Bukhaari (2215). 

Fourthly: 

Something else that will convince you that the previous answer is correct is if you were to imagine that all your friends borrowed things from you, each of them borrowing one of your possessions, whether it is clothes or other items or books, then none of them returned what they had borrowed from you on the grounds that a long time had passed, and the borrowed item had suffered some kind of wear and tear. Would you be happy with that? Would you think that that was an appropriate kind of interaction between people? Would you not be upset and annoyed, especially if the borrowed item was something that was dear to you or was somewhat prcious? Perhaps then you will understand that this world cannot be in good order unless people pay heed to morals and ethics, and render trusts back to those to whom they are due. This is an important principle of human co-existence. “Verily! Allah commands that you should render back the trusts to those, to whom they are due; and that when you judge between men, you judge with justice. Verily, how excellent is the teaching which He (Allah) gives you! Truly, Allah is Ever All-Hearer, All-Seer” [an-Nisa’ 4:58]. 

The idea is quite clear: there is no room for trickery and we should aspire to reach the highest level of honesty with everyone. 

The Prophet (blessings and peace of Allah be upon him) said: “Render back the trust to the one who entrusted it to you, and do not betray the one who betrayed you.”

Narrated by Abu Dawood in as-Sunan (3534); classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

And Allah knows best.