Sayyidah Aishah (may Allah be pleased with her) said: the Messenger of Allah (peace and blessings be upon him) woke up and the time of the prayer had come. Water was searched for but it was not found, so the verse of tayammum was revealed.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 1102
It is narrated from Nafi' (may Allah have mercy on him) that Sayyiduna Ibn Umar (may Allah be pleased with him) performed tayammum at a place called Mirbad al-Na'am and prayed, and that place is three miles from Madinah. Then he entered Madinah while the sun was high, and he did not repeat the prayer.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 1103
Ibn Muslim (may Allah have mercy on him) said: Abu Amr al-Awza'i (may Allah have mercy on him) was asked: "The time of the prayer has come and the water is away from the road; is it obligatory on me to turn aside from the road and go to it?" He said: Musa bin Yasar (may Allah have mercy on him) narrated to me from Nafi' (may Allah have mercy on him), and Nafi' (may Allah have mercy on him) from Ibn Umar (may Allah be pleased with him), that he would be on a journey and the time of the prayer would come while water was one or two bow-shots (ghalwah) away from him, or about that, and then he would not turn aside to it.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 1104
Ruzayq bin Hakim (may Allah have mercy on him) narrates from his father (may Allah have mercy on him) that he said: I asked Sa'id bin al-Musayyab (may Allah have mercy on him) about a shepherd with his sheep, or such a shepherd, who becomes junub (in a state of major ritual impurity) and between him and the water there is a distance of two or three miles: what should he do? Ibn al-Musayyab (may Allah have mercy on him) said: "Such a person should perform tayammum with clean earth."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 1105
It is narrated from Sayyiduna Ali (may Allah be pleased with him) that he said: "Keep searching for water until the last part of the time; if you do not find water, perform tayammum and then pray."
This is not authentic from Ali.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 1106