We tend to forget that our opponent has a lot to say on when the fight stops. I've seen people dead from small (.22, .32) rounds and I've seen people still aggressive after receiving multiple 7.62 hits. While projectile design has come a long way in recent years, CNS hits with bullets big enough to cause immediate disruption are going to be the determining factor. While I wouldn't feel undergunned with a 9MM or a .38, and do carry them on occasion, I personally wouldn't carry anything smaller.
Penetration comes into play also. Your weapon needs to be able to drive the bullet into your aggressor far enough to reach the CNS (spine). No problem if you're shooting someone in the back, but from the front or side, which is the normal aspect presented by an aggressor, you may have to cut through several inches of clothing, muscle, organs, etc. Shots to the vital organs will eventually take your aggressor out of the fight (the old blood out - air in depressurize the body method) but that may take some time. Your opponent doesn't need much time to bring a weapon to bear and pull the trigger.
My EDC is a Springfield doublestack 1911 in .45., primarily because I've carried a 1911 for 20+ years and it's what I'm used to. My recommendation for new shooters is to carry the biggest gun they can control for effective follow on shots, but small enough that they will actually carry it. And herein lies the trade-off we're forced to make. I'm 6'1", 190#, and with experience in picking the proper holster and clothing can conceal my full size 1911. There are occasions where my wardrobe isn't conducive to carrying the full size gun, so a Kahr PM9 fills the leather. I'm more comfortable with the 1911, as it fills my hand, has the benefit of a longer sighting radius, and is punching nearly half inch holes on target. But I'm equally confident in my abilities to defend myself with the Kahr. I understand my shot placement is going to be slightly more critical with the 9MM. But it will do the job. For a 5' tall female that weighs 100#, hiding a full size handgun will be problematic. An easily concealed .45 may be difficult for them to effectively handle. So they may have to make the trade-off to a lighter caliber weapon, giving up muzzle energy for control. There are many factors involved in what you can effectively carry, and a reasoned approach to a decision is necessary.
If all you have is a .22, it's better than nothing. However, there currently exists a huge selection of easily concealable, controllable weapons in proper defensive calibers so that CWP holders can provide themselves with a much better chance of survival.