which martial art would be best for a nine year old girl?

Good Classical Judo.

It teaches how to use leverage to defend yourself. Also she will learn how to fall without hurting herself - something very important since often times the untrained can fall or be thrown.

Edit:

The person that suggested BJJ is probably 10 years old and never trained in his life.

I think old school kodokan judo or bjj is great for young girls. Also try googling verbal judo I bet that would come in handy for bitchy highschool chicks. Judo is one of the best art I've seen or heard about but all arts are good and serve some purpose I'd find something that she enjoys even if that's tkd or tiggy karate and let her stick with it. Keep in mind though the only real full contact arts for kids are grappling arts and if you want her to be able to defend herself you must look into full contact arts.

This is a question that is quite frequent in this place and there is only one answer to that question, we cannot help you. In the end, all of us will only tell you that our style is the best. I will tell you that any martial art will work. I suggest you look around and do trials. Ask your daughter which style she likes best and then make the choice. Most of this is because I believe each martial artist has a preferred fighting style, be it grappling, striking, hybrid(grappling and striking), or weapons focus. Therefore, use this site only as a reference and make your own choice. My list of suggestions are:

Striking:

Taekwondo WTF or ITF(since your in the UK/Ireland)

Taekkyeon(Korean dancelike art, father of TKD, not sure if you'll find someone teaching it there)

Tang Soo Do

Muay Thai

Kickboxing

Capoeira(has dancing and gymnastic elements good for girls)

Savate(French Kickboxing)

Wing Chun

Shaolin Kung Fu

Hung Gar

Taichiquan

Baguaquan/Piguaquan(usually both arts are taught together)

Other Kung Fu styles

Shotokan Karate(Father of Taekwondo)

Kyokushin Karate

Other Karate styles

Grappling

Brazilian Jiujitsu

Japanese Jiujitsu

Judo

Aikido

Hybrid

Jeet Kune Do(Bruce Lee made it)

Chun Kuk Do(Chuck Norris made it)

Hapkido

Hwa Rang Do

Kuk Sool Won

Pencak Silat

Other Silat styles

Weapons

Kumdo(Korean)/Kendo(Japanese)(Japanese is more focused on the meditation and Korean is more focused on the sword)

Kali/Escrima/Arnis (Filipino Martial Arts, almost no difference)

Fencing

I started martial arts in my 20s, I wish I could have started earlier, but I cannot time travel. If you want to take Karate, then go for it. I know plenty of girls that train and can hold their own. You are never too old.

All are good, just find something most convenient for you to drop and pick her up.

Thats what messes up people in the long run.

I would say Judo is girls favourite otherwise, but all martial arts are great for body and mind.