Murders and false claims of Prophethood arise
A force was being prepared under Usama Ibn Zaid, whose father was killed at Muta, against the Byzantines, to exact the long-delayed reparation for the murder of the envoy to Syria. However, the news of the Prophet's (ﷺ) sickness and failing health caused that expedition to be stopped. This news was soon noised abroad and produced disorder in some districts.

Three pretenders had arisen who gave themselves out as prophets and tried by all kinds of imposture to win over their tribes. The most dangerous of these pretenders was known as Al Aswad. He was a chief of Yemen and a conjurer. He soon succeeded in gaining over his tribesmen and, with the help, reduced to subjection many of the neighboring towns. He killed Shahr, whom the Prophet(ﷺ) had appointed as Governor of Sana in the place of his father Bazan, who had just died. Bazan had been the viceroy of Yemen under Chosroes of Persia; after he had adopted Islam he was allowed by the Prophet(ﷺ) to remain as Governor of Yemen. He was able to convert to Islam all the Persian colony in that province. Al-Aswad, the conjurer, had now killed Shahr, but soon after he was massacred by the Persians of Yemen.

The other two pretenders, Tulayha and Haroun by name, were not suppressed until after the death of the Prophet, during the reign of Abu Bakr. Haroun, better known as Mussaylamah, addressed to the Prophet(ﷺ) a letter which ran as follows:
"From Mussaylamah the Prophet of Allah, to Muhammad the Prophet of Allah. Peace be to you. I am your partner. Let the exercise of authority be divided between us. Half the earth will be mine, and half will belong to your Quraish. But the Quraish are too greedy to be satisfied with a just division."

To this letter the Prophet(ﷺ) replied as follows:
"From Muhammad the Messenger of Allah to Mussaylamah the liar. Peace be to those who follow the right path. The earth belongs to Allah. It is He Who makes the reign whomsoever He pleases. Only those will prosper who fear the Lord."

Prophet Muhammad's last days
The health of the Prophet grew worse. His last days were remarkable for the calmness and serenity of his mind. He was able, though weak and feeble, to lead the public prayers until within three days of his death. He requested that he might be permitted to stay at 'Aisha's house close to the mosque during his illness, an arrangement to which his other wives assented. As long as his strength lasted, he took part in the public prayers.

The last time he appeared in the mosque he addressed the congregation, after the usual prayers were over, in the following words:
"O Muslims, if I have wronged anyone of you, here I am to answer for it; if I owe anything to anyone, all I may happen to possess belongs to you."

A man in the crowd rose and claimed three Dhirhams which he had given to a poor man at the request of the Prophet. They were immediately paid back with these words:
"Better to blush in this world than in the next."

The Prophet then prayed and implored Allah's mercy for those who had fallen in the persecution of their enemies. He recommended to all his followers the observance of religious duties and the leading of a life of peace and goodwill. Then he spoke with emotion and with a voice still so powerful as to reach beyond the outer doors of the mosque:
"By the Lord in Whose hand lies the soul of Muhammad as to myself, no man can lay hold on me in any matter; I have not made lawful anything excepting what Allah has made lawful; nor have I prohibited anything but that which Allah in His Book has prohibited."[

Then turning to the women who sat close by, he exclaimed:
"O Fatimah, my, daughter, and Safia, my aunt, work you both that which procure you acceptance with the Lord, for verily I have no power to save you in any wise."

He then rose and re-entered the house of Aisha.

The death of Prophet Muhammad(ﷺ), peace be upon him
After this, the Prophet never appeared at public prayers. A few hours after he returned from the mosque, the Prophet(ﷺ) died while laying his head on the bosom of Aisha. As soon as the Prophet's (ﷺ) death was announced, a crowd of people gathered at the door of the house of Aisha, exclaiming:
"How can our messenger be dead?"

Umar said: "No, he is not dead; he will be restored to us, and those are traitors to the cause of Islam who say he is dead. If they say so let them be cut in pieces."

But Abu Bakr entered the house at this moment, and after he had touched the body of the Prophet(ﷺ) with a demonstration of profound affection, he appear at the door and addressed the crowd with the following speech:
"O Muslims, if anyone of you has been worshipping Muhammad(ﷺ), then let me tell you that Muhammad(ﷺ) is dead. But if you really do worship Allah then know that Allah is living and will never die. Do you forget the verse in the Quran (Ch 3:144 Holy Quran):

وَمَا مُحَمَّدٌ إِلَّا رَسُولٌۭ قَدْ خَلَتْ مِن قَبْلِهِ ٱلرُّسُلُ ۚ أَفَإِي۟ن مَّاتَ أَوْ قُتِلَ ٱنقَلَبْتُمْ عَلَىٰٓ أَعْقَـٰبِكُمْ ۚ وَمَن يَنقَلِبْ عَلَىٰ عَقِبَيْهِ فَلَن يَضُرَّ ٱللَّهَ شَيْـًۭٔا ۗ وَسَيَجْزِى ٱللَّهُ ٱلشَّـٰكِرِينَ ١٤٤

Muḥammad is no more than a messenger; other messengers have gone before him. If he were to die or to be killed, would you regress into disbelief? Those who do so will not harm Allah whatsoever. And Allah will reward those who are grateful.

Upon hearing this speech of Abu Bakr(رضي الله عنه), 'Umar(رضي الله عنه) acknowledged his error, and the crowd was satisfied and dispersed.

Al-Abbas, the Prophet's uncle, presided at the preparation for the burial, and the body was duly washed and perfumed. There was some dispute between the Quraish and the Ansars as to the place of burial; however, Abu Bakr(رضي الله عنه) settled the dispute by affirming that he had heard the Prophet say that a prophet should be buried at the very spot where he died.

A grave was accordingly dug in the ground within the house of Aisha and under the bed on which the Prophet died. In this grave the body was buried, and the usual rites were performed by those who were present. Thus ended the glorious life of that Prophet Muhammad (ﷺ).

May the peace and blessings of Allah be upon him.

Reference: From "Stories of the Prophets" by Al-Imam ibn Kathir