.22 vs .380 for concealed carry self defense?

Hi all, I took a firearm safety course last night and the instructor had me shooting a S&W .22 Compact. It fit great in my hands, and shot well. It was my first time using a handgun, and it seems to be a good fit, given the accuracy I was able to nail down in just 40 rounds at 15 feet.

The instructor said he thinks this would be a great choice for me to concealed carry as self defense, but my dad and brother said I was firing a toy gun, that .22s are unreliable with a tendency to misfire, and they won't incapacitate someone. Instructor said a 9mm is not necessary for concealed, and to not let anyone talk me into it. Dad and brother want me to go back and try a .380, which they said is a step down from a 9mm.

Anyone care to share any insight? For reference, I'm 28 years old, 5' tall, 100lbs, with baby hands, so I definitely need recommendations for guns that have a better grip for smaller hands.

As a total newbie, any and every tip is more than welcome! Thank you!!