I will not get it done with a gun. It takes ages to heal, and there is a high risk of it getting infected/scaring in an ugly manner.
If you decide to get her ears pierced at 11, go to a proper piercing specialist, it will cost more, but this her body we are talking about. If it goes wrong, she can be scarred for life.
I had my ears pierced several times with guns as I grew up, had infections in them, and now if I wear earings they become red and sore.
I had my nose pierced with a gun and it became infected and sore for a long time, had to take it out - this was done professionally and kept clean.
I had my nose pierced again, but this time with a needle by a professional piercer, and my lip peirced twice by a professional piercer, the healing time was signifantly quicker, it was much less painful afterwards.
A gun shocks the skin and increases the risk of it scaring to protect the site. A needle is slower and give the skin time to cope with the trauma and this healing time/scaring is reduced.
Of course I am not suggesting nose/lip piercings, and I personally think 11 is too young, but the principle of a needle is much cleaner, and it hurts no more/no less (in fact I thought less as there was no sudden jerk of a gun).