It seems that the self-timer select switch is always "on". It may be dirt or a bronze spring that needs alignment, or (worst) an electronic problem (this seems unlikely) If some gentle tapping near the self-timer lever does not solve the problem, you'll have to open the camera to check the self-timer switch, or send it for repair.
If you decide to DIY, there is an XA repair manual here
Taking out the bottom plate is not hard (it is four screws plus the one below the self-timer and battery check level). You'll probably have to take out the sliding door as well. Beware of the sliding door roller, it is easy to lose.
I attempted repair on my XA2 on a harder problem and decided not to complete work, but documented the steps I took here