I think it is wonderful that your child is showing an interest in playing an instrument and making music. You should certainly encourage that desire in any way possible.
There are many very good quality smaller size guitars that might be a good fit for a six year old. In order to increase the chances that your child will actually find the experience rewarding enough to stick with it as she gets older, you should get her the best quality smaller size guitar you can afford. Avoid the ones that are sold in the toy department at Walmart.
Name Brand Guitar makers such as Yamaya, Taylor and Martin - all make guitars that are smaller scale. Sometimes they call them travel guitars and they can certainly be used that way. I personally love the Baby Martins because they sound awesome for such a small guitar. Taylor makes the Baby Taylor and Martin makes several versions of their Little Martin such as the LX1. Yamaha makes the Yamaha Junior.
To make the smaller guitar even easier on tender fingers you might consider restringing it with a custom finger friendly string set like the one described here. Custom Finger Friendly String Set for Beginning Guitarist
Of course a ukulele is easier to play and easier to learn. But it is not a guitar. I own a ukulele but rarely play it because I personally prefer the way my guitars sound. My uke is a solid mahogany baritone uke but still sounds thin compared to even my Taylor GS Mini Travel Guitar.
But since ukuleles can be very inexpensive, and even the lower priced ones are relatively easy to play, you could get both a pair of ukes and a good quality short scale guitar. If your child gives up on the guitar, you will have a travel guitar to take on vacation.
Enjoy your time making music together with your child.