Hi C.L,
Just a few questions on your setup to help me understand the problem. You mentioned that you are using the part in a unity gain configuration. When you see the supply dropping to +/-5V, are you already applying a signal to the input of the op amp? Does the same behavior follow for the isolated supply without an input signal applied? Also, are you configuring the part as in Figure 12 of the datasheet?
Thanks!
Kris