I say, and with God's success: The fear of cold is a valid excuse for performing tayammum instead of ghusl, if a resident or traveler fears illness from using water for ghusl, or the loss of a limb, or death; because the lack of water and warmth, even if rare, does not contradict the permissibility of tayammum. Tayammum due to cold is specific to ghusl, not to wudu, according to the correct opinion. From Amr ibn al-Aas : "I had a wet dream on a cold night during the battle of 'Dhat al-Salasil', and I feared that if I performed ghusl I would perish, so I performed tayammum and then prayed Fajr with my companions. They mentioned this to the Prophet , who said: 'O Amr, did you pray with your companions while you were in a state of janabah?' I informed him of what prevented me from performing ghusl, hoping that I heard that Allah says: {And do not kill yourselves} (An-Nisa: 29), and the Messenger of Allah  laughed." This is reported in Al-Mustadrak 1: 285, Al-Sunan Al-Sughra 1: 185, Al-Sunan Al-Bayhaqi Al-Kabir 1: 225, Al-Sunan Al-Daraqutni 1: 178, Al-Sunan Abu Dawood 1: 92, and Musnad Ahmad 4: 203. Refer to: Fath Bab Al-I'naaya 1: 110, and Allah knows best.