On my 22 caliber rimfire handguns, I use Mobil 1 10w-40 synthetic motor oil.
Several years ago I was shooting a new 325 round value pack thru my Glock 44. I was experiencing at least 2 or 3 stove pipes per magazine. The pistol was freshly cleaned and lubed with Ballistol which I still use on fire control parts. After a little over 100 rounds I gave up. Afterwards while cleaning the pistol, I decided to try the Mobile 1 on the Glock. I’ve been pleased with using it on my center fire pistols. So, after the cleaning and lube, I fired the remaining 200 or more rounds thru the Glock 44 without the first issue.
I have not tried grease on a rimfire pistol. I wonder if grease being thicker if that would possibly slow the slide down enough to cause malfunctions. Maybe not. I do use a thin layer of Mobil 1 synthetic grease topped off with a couple of drops of oil on the rails of my center fire pistols. That combination works for me and the pistol stays lubed a very long time.