This is not uncommon with wrist straps which is why neither I nor Dr. Clark recommends them.
The first problem is that many wrist straps use either stainless steel or nickel plated brass.
In either case, the contact will leach nickel which is classified as a tissue irritant. This can enter the skin and provide long term sensitivity.
The second problem is that the wrist straps provide only a small contact in one of the more sensitive areas of the body. The small contact accentuates the problem of skin sensitivity and the use of plain water can aggravate this even more. Even worse is distilled water. Salt water works better but will still allow some irritation. Conductive gels can help some but do not entirely eliminate all of the problems.
For the absolute best results, hold copper pipe paddles in the hands and place your bare feet on augmentation copper footpads. Cover both the copper paddles and the footpads with a good quality paper towel that has been soaked in salt water ( about 1 tsp to 6 oz of water ). You will get little irritation and about 4 to 5 times the effectiveness.