In a situation where continued dispute could lead to serious consequences (like a very treated argument) and a promise or lie is made to avoid disaster, must it be fulfilled?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Reconciling people is one of the acts by which a person gets near to Allah. Allah says: " There is no good in most of their secret talks save (in) him who orders Sadaqah (charity in Allah's Cause), or Ma'rûf (Islamic Monotheism and all the good and righteous deeds which Allah has ordained), or conciliation between mankind, and he who does this, seeking the good Pleasure of Allah, We shall give him a great reward. (An-Nisa 4:114)

Due to the importance of reconciliation, the Prophet (Sallallahu Alaihi wa Sallam) allowed telling a lie for this purpose. Imam Al-Bukhari and Muslim reported that Um Khulthum, may Allah be pleased with her, said: "I heard the Prophet (Sallallahu Alaihi wa Sallam) saying: "The liar is not the one who reconciles between people by saying good or reporting good."

Ibn Al-Jawzi said: "Its criterion is: lying to achieve any praised aim that is not achieved except by telling a lie, is permissible if the aim is permissible, and if the purpose is an obligation, then it is an obligation to lie."

But when reconciling people, you should not promise to pay a debt that you are unable to pay. So, if you took the responsibility of being a guarantor of someone and promised to pay his debts while you are unable to do so, there are two cases: either the creditor acknowledges that, and you told him that "I am a guarantor", "I am a sponsor," or "I am responsible for what he has to pay", then you have to fulfill what you promised. And the creditor has the right to ask you about the money. However, if you mentioned something which is not straight forward and the meaning is that you will fulfill whatever is needed, so you have to fulfill your promise as well. For instance, if the creditor wants to put the debtor in prison and you told him 'I will pay the money', so you meant 'I am his guarantor, and do not put him in prison', in this case you have to fulfill the guarantee, even if the wording 'I will pay the money' is not straightforward, but in this circumstance, it is.

The second case is if you promised to pay someone's debt the other party is asking of him, then you are not obliged to fulfill this guaranty, because the guaranty can be correct only if what is guaranteed is acknowledged as being a debt by the debtor. It is the same case if the matter is just a promise, then it is not an obligation to fulfill it, but it is desirable.

What is mentioned above  is what the scholars, may Allah have mercy on them, said, like Zakariya Al-Ansari- in Sharh Al-Manhaj' and Al-Bahuti in Keshaf Al-Qina, and others.

Allah knows best.