Fullfilment of covenant, agreement, oath and given word
Please don't bring cases from life, because in life everything takes another turn. Let's see what the law says. And this law is valid not only between Muslims but applies to all people in our interpretation.
O ye who believe! fulfil (all) obligations… (Quran 5:1)
If Muslims are charged with the abusing of their agreements and obligations here you are the clear evidence that Quran prescribes to be faithful for promises and men have to keep their deals and agreements. If we Muslims are not doing it properly, we commit the same error, when other people are unreliable no matter who follow which Law or Book. The mistake is always there in the man and not in the teachings or Books.
…and fulfil (every) engagement, for (every) engagement will be enquired into (on the Day of Reckoning) (Quran 17:34)
… to fulfil the contracts which ye have made (Quran 2:177
Those who fulfil the covenant of Allah and fail not in their plighted word; (Quran 13:20)
Nay.- Those that keep their plighted faith and act aright,-verily Allah loves those who act aright. (Quran 3:76)
Kathir bin 'Amr bin 'Awf Al-Muzani narrated from his father, from his grandfather, that the Messenger of Allah (peace be upon him) said: "Reconciliation is allowed among the Muslims, except for reconciliation that makes the lawful unlawful, or the unlawful lawful. And the Muslims will be held to their conditions, except the conditions that make the lawful unlawful, or the unlawful lawful."
Grade: Sahih
Reference: Jami` at-Tirmidhi 1352
Fulfil the Covenant of Allah when ye have entered into it, and break not your oaths after ye have confirmed them; indeed ye have made Allah your surety; for Allah knoweth all that ye do. (Quran 16:91)
The immediate reference may or may not be to the oath of fidelity to the Prophet taken at 'Aqaba fourteen months before the Hijrah and repeated a little later: see 5:7. But the general meaning is much wider. And this may be viewed in two aspects; - Every oath taken, or covenant made, is a Covenant before Allah, and should be faithfully observed. In this it approaches in meaning to 5:1. - In particular, every Muslim makes, by the profession, of his Faith, a Covenant with Allah, and he confirms that Covenant every time he repeats that profession.
Further ayas about that subject:
17:34, 26:181-182.
Contracts on business, property, marriage, or any other oath, agreement and covenant is made before Allah (SWT). No one can avoid His presence, and that is the basis of the thesis. So, a contract with anyone, no matter he/she is a Muslim or non-Muslim, is considered an agreement with Allah.
Regardless of who is the subject of the covenant, the agreement is a covenant with Allah. Therefore, non-observance of it is a violation of an agreement made with Allah.
So, the rumor that a word given to an unbeliever is not obligatory is not true because any given word is a word given to Allah.