Fiqh-us-Sunnah Volume 1, Purification and Prayer, Fiqh 1.068.

Whoever cannot get water or soil may pray in whatever state he is in, and he will not have to repeat his prayer later. This is based on what Muslim related from ‘Aishah. She had borrowed some jewellery from ‘Asma and it broke (and fell). The Messenger of Allah, upon whom be peace, sent some people to search for it. The prayer time came and they had to pray without ablution. When they came to the Prophet, they complained to him and the verses of tayammum were revealed. Usaid ibn Huzhair said, “May Allah give you good recompense. Allah never reveals an order with respect to you except that He removes by it some hardship and gives the Muslims some benefit.” The companions prayed while in a state of impurity, but the Prophet did not admonish them nor did he order them to repeat their prayers. Says an-Nawawi, “That is the strongest statement of proof (on this question).”