I don't know how much truth there is to this, but my dad is friends with a detective. I think his service weapon is a 9mm. His personal weapon that he keeps on him at all times is a 22. He says that if he has pants on you can be sure he also has his 22. He says that he knows of a case where a guy was chasing someone. The guy being chased had a 45 or 40 or 9mm (I forget which). He shot the chaser five times and hit him each time, but the guy kept coming. He said the shooter didn't hit any vital organs and the rounds went through the guy. The detective's thoughts are if you hit someone five times with a 22 at least one of the rounds is going to bounce around inside the body and mess something up.
You guys are the weapons experts. Any truth to it?
"The bullet will bounce around" is a common canard about the .22 LR. The bullets don't bounce around, the physics of it don't work that way. They may corkscrew or tumble through tissue, or be deflected by bone, but they will not reverse direction. Instead, the bullet will transfer its kinetic energy into the tissue.
Handgun don't really work like you see in the movies. A pistol bullet will not pick up a man and throw him backwards off his feet- if it did, it would throw the shooter back with equal force. For every action... Also, you can't count on an instantaneous stop from a brain or spinal cord hit during a gunfight. So, the three things a handgun bullet must do are:
penetrate sufficiently to damage major organs and blood vessels; generate a permanent wound channel sufficiently large for blood to exit the organs and the body rapidly; and cause enough pain to distract and dispirit your assailant. Pain is a result of the energy transfer from bullet to flesh, and bullets which pass
through the body do not transfer all of their energy into the target.
A .22 LR bullet may or may not expand, depending on the bullet shape, velocity, and whether or not it strikes bone while passing through tissue. Even if it doesn't expand, it still makes a 1/4 inch permanent wound channel and hurts like hell- two of the factors of handgun effectiveness. As I have said before, four .22s in the chest, then four in the face, will change the balance of power in any deadly force confrontation.
9mm ball (full metal jacket) and light jacketed hollowpoints are often moving
too fast to expand inside the body. They hold their shape and generate a 3/8 inch diameter permanent wound channel through both sides of the body, but are less painful because they did not transfer all of their energy to the target. Better terminal performance in 9mm would come from the heavier, slower bullet, particularly the 147gr SJHP. An exception is the new Critical Defense round from Hornady, which is built to reliably expand from .355 inch to more than half an inch in diameter, causing a larger permanent wound channel and greater energy transfer.
.357, .40, .44, and .45 have almost identical success rates against human targets, for the same reason. These bullets penetrate deep into the torso, reliably expand and make a larger permanent wound channel, and transfer more energy into the target, causing greater pain. The tradeoff is slower rate of fire because of longer recovery time- except of course for my .45, which is as fast as lightning

So, what does it all mean? Remember the first rule of gunfighting:
Bring a gun. A .22 in the hand beats a .45 in the gun safe.
Carry and shoot what you are comfortable with. A hit with a .22 beats a miss with a .44. Like DAT says,
Anything worth shooting is worth shooting at least twice. If you carry a small caliber pistol (.22-.380) into a deadly force confrontation,
shoot more. 4 to the chest, 4 to the face. Reload, and be prepared to continue the fight.
Note: I carry Hornady Critical Defense ammunition in all of my handguns, (.45 ACP, .357 Magnum, and .38 SPL +P) and recommend it without reservation. This is not a paid endorsement, although I wish it was.