We are using SAM Broadcaster Pro.

We play programs from the Pacifica Network in addition to playing music. What happens is that a program will be running and suddenly, a music file will start playing interrupting the program. I manually put the program back in the queue and restart the program at the point where it was interrupted.
 
It seemed to me that the Gap Kill had something to do with this, so I would disable the Gap Kill for that particular program. That seemed to work at first but then it continued, so I disabled Gap Kill for SAM entirely. The problem still occurs, so now I don’t know what to do.

Looks like a wrong / forgotten entry in the "Event Scheduler" or simply an active PAL script. So I recommend to check both Event Scheduler and PAL scripts. As Gap Killer is disabled I would make an alternative check of cross fading settings in general settings and in track settings (if applicable).