Make sure you hold it down and keep the pilot lit long enough before releasing button
I had a problem with my Little Buddy where the pilot would go out when I released the On button. Replaced the thermocouple and it worked temporarily. I finally realized that the OFF button was sticking. A couple of shots of WD40 silicon spray loosened it up and now the Little Buddy works fine. Never tried this on the larger models with the twist knob, but will the next time I have a problem with one of them.