The time limit given for Muslims to appoint a Khalifah is two nights. So no Muslim is allowed to stay for more than two nights without having a bay’ah on his neck. Making the upper limit as two nights is due to the fact that the appointment of a Khalifah is fard from the moment the previous Khalifah dies or is deposed. But it is allowed to delay the appointment while engaging in it for two nights. If the delay exceeded two nights and the Muslims did not establish a Khalifah, the matter is examined. If the Muslims were busy in establishing a Khalifah but could not complete it within two nights due to overpowering matters they couldn't resist, then the sin will drop from them because they are engaged in establishing the duty and they were forced to delay it by compelling power. The Prophet (saw) said:

But if they were not engaged in performing the duty, then they will all be sinful until the Khalifah is appointed and only after that the fard will drop from them. But the sin they committed in neglecting the establishment of a Khalifah does not drop from them, it rather remains on them, and Allah (swt) will bring them to account for it the same way He (swt) brings any Muslim to account for any disobedience he commits when he neglects to perform a duty.

As for the evidence concerning the two night's time limit given to Muslims to perform the duty of establishing a Khalifah, it is the Ijma’a of the Sahabah. The Sahabah (ra) met in the courtyard of Banu Sa‘ida, to discuss the appointment of a successor to the Messenger of Allah (saw) as soon as the news of the death of the Prophet (saw) reached them. They kept discussing in the courtyard, and on the second day they gathered the people in the mosque in order to give the bay’ah. This consumed two nights and three days. In addition, when ‘Umar (ra) became certain that his death was imminent as a result of the stab wound, he entrusted the people of consultation (shura) and gave them three days to choose a new Khalifah. He recommended that if after the three days an agreement was reached about a Khalifah then the dissenter should be killed after the aforementioned three days. ‘Umar (ra) also empowered fifty Muslims to carry out this action i.e. to kill the dissenter, despite the fact they were from the people of shura and of the eminent Sahabah (ra). This order was given in the presence of the Sahabah (ra), and no one was reported to deny or disagree with it, so it becomes Ijma’a of the Sahabah (ra) that Muslims are not permitted to stay without a Khalifah for more than two nights and three days, and the Ijma’a of the Sahabah is a legitimate Shari’ah evidence (daleel) like the Qur’an and Sunnah of the Messenger of Allah (saw).