Which instrument for a 6yo?

[Hulababy]

DD is really keen to start some form of music lessons. She will have the chance for school based lessons from next academic year. However, many of her school friends are starting this year outside of school, and this is something DD has always been keen to do.

She would love to learn to play a piano - but they are so expensive, even second hand ones in shops seem to be. They are also big!
She has a big keyboard and will play about with thatm and we have recorders but she isn't as bothered about that at present.
So what else could we have a look at for her; something smaller maybe?
Any musicians out there who could recommend a good first instrument?
Also, where to go to look and buy?
And how to find a decent music teacher?

DD started clarinet at the beginning of this academic year and is already playing tunes - I am so proud of her - she is small for her age but a clarinet is handleable for her and it isn't that expensive as we hire it from our local music services - about £30 a term I think although that is just until we know she wants to carry it on - they are about £200 - £300 to buy.

Really agree with the advice to let her play whatever instrument she wants to play, that is the most important thing.
If it is the piano, see if you can find a decent one second hand, through a reputable dealer.
Recorder is also a good starter instrument.

My ds (4) has started the piano at school and loves it (tried to get him to learn the recorder but he wasn't at all interested). He was adamant that he wanted to learn the piano and did a test at school to see if he was ready to learn. We got a Yamaha digital piano and piano stool for under £500 from an online shop (next day delivery too). The teacher recommended a digital piano to start with and it sounds like a normal piano but can also sound like strings/organ etc and it has about 50 preset tunes so you can hear how good it can sound. My ds loves practising and also playing along to the preset tunes (well trying at least). It also has headphones and a volume control (perfect for 6am practice sessions!!).