Yep, the 120p is a great little light, but it does have a few quirks.
While you've got the light apart getting ready to clean, check the retaining ring in the head and the tail cap. The retaining ring is what holds the tailcap switch in place and the head in place. It's the very thin silver disc at the bottom, inside the tailcap (and head), that has four small notches across from each other. Use a pair of tweezers (or other suitable tool) to tighten that disc clockwise. Same thing in the head, where the retaining ring holds the head unit in place.
Once you've tightened those, you'll want to check the main spring (the one that wraps around the battery and conducts controls signals from the tailcap to the head). It might need to be stretched a bit, but don't pull on the ends to stretch it, as this will make the diameter smaller and it might stick on the battery. You want to gently "unroll" it just a bit, which you do by grabbing one end on the left hand and one end in the right hand, with your fingers around the outside of the coil, then gently twist them counterclockwise (the end in the left hand twist away from you, the end in the right hand, twist toward you). Not too much, just a bit.
Ok, you're ready to clean. Electrical contact cleaner is best, but if you don't have that, isopropyl alcohol will work in a pinch, along with some Q-tips cotton swabs. Start with the tailcap. Just a bit of cleaner on the Q-tip, then get it down inside to clean the retaining ring you just tightened and that silver area just below that, as well as the tailcap spring.
Next the head. It's very important to hold the head up when you do this (with the opening facing down), as you don't want cleaner to run down inside where the electronics are. Clean around the edges where the tubes make contacts, and the silver area where the spring makes contact, and where the battery makes contact. Next clean both ends of the main spring and both ends of the battery.
Now, reassemble with a fresh battery and tighten all parts well. That should do it. If not, consider a full reset and try again. Good luck.