Yahya related to me from Malik from Nafi that Abdullah ibn Umar used to do tayammum up to his elbows.
Malik was asked about how tayammum was done and what parts were covered and he said, "Strike the ground once for the face and once for the arms and wipe them to the elbows."

وَحَدَّثَنِي عَنْ مَالِكٍ، عَنْ نَافِعٍ، أَنَّ عَبْدَ اللَّهِ بْنَ عُمَرَ، كَانَ يَتَيَمَّمُ إِلَى الْمِرْفَقَيْنِ . وَسُئِلَ مَالِكٌ كَيْفَ التَّيَمُّمُ وَأَيْنَ يَبْلُغُ بِهِ فَقَالَ يَضْرِبُ ضَرْبَةً لِلْوَجْهِ وَضَرْبَةً لِلْيَدَيْنِ وَيَمْسَحُهُمَا إِلَى الْمِرْفَقَيْنِ .