Is it permissible for a sick person who fears harm from using water to perform tayammum in lieu of ghusl and for every prayer?

If you fear harm from performing ghusl (ritual bath) to remove janabah (state of ritual impurity), then it is permissible for you to perform tayammum (dry ablution) with earth, because water is considered non-existent in such a case. This is supported by the tayammum performed by Amr ibn al-Aas (may Allah be pleased with him) due to extreme cold and the Prophet's (peace be upon him) approval of his action, as he cited Allah's saying: (And do not kill yourselves. Indeed, Allah is to you ever Merciful). [Quran 4:29] And "pure earth" (الصعيد الطيب) refers to dust/earth, and this is the most preponderant opinion. If the rock you performed tayammum with had dust clinging to it, then your tayammum is valid. Otherwise, you must repeat the prayers you performed.