First, try and see if you can get the 6 year old to think about "what if there are colors we can't see"? Explain to her that the color we see is the color of "light".
Now, show her a remote control, and press some button. There's an IR bulb up front, ask her if it flashes when you press the button (it shouldn't). Now, use a phone camera to look at the IR bulb, most phone cameras will show white light when the button is pressed. Explain to her that the light coming from the remote is "invisible", in the sense that it's of a color we can't see. However, the camera can see it because the camera sees slightly more "colors" than we can, and when it tries to display it it shows it as white.
Explain to her that this is "infrared" light, a light that is "more red than red itself". Whenever someone turns on the TV, a light signal is sent to the TV. (You may want to explain that this light has some "bending" capabilities, but that's not entirely necessary).
This ought to get her past the mental block when it comes to "light that isn't light". Mentioning that some animals see more/less colors than we do helps.
Now, talk about the spectrum:
Explain that the light that we can see is a very small portion of the kinds of light that actually exist. The spectrum is what she sees when she looks at a rainbow, but it really doesn't "stop" at red or purple; she just can't see it.
If you wish, you can then talk about radio waves, and how they are light that can easily "bend" (i.e. diffract). Talk about X-rays, which is light that can pass through skin but not bones. This can actually lead to an interesting side track where you explain how an X-ray is nothing but a photograph with a different kind of light.
Once you reach here, it's easy to explain UV. Mention that while the sun emits a lot of visible light, it's not limited to the visible spectrum and emits a significant amount (much less, but not negligible) of UV and IR as well.
You can actually extend this to sound as well, talk about how there are sounds we can't hear. For that matter, sounds just outside your hearing range will be clearly audible to most six year olds. If you can generate increasing frequencies from your computer (It's actually possible for our vocal cords to work in the inaudible ranges, but it takes some practice to get that to work so it's just easier to use a computer), you can both show here that different people/ages have different frequency ranges1, and that there are sounds that even she can't hear. (to do the latter you may want to set up a microphone and have it show the amplitude on the screen or something). Similarly, you can go to lower frequencies (and show the transition from invisibly fast vibrations but audible sounds to visible vibrations and inaudible sounds in a string instrument or possibly a rubber band). It's a good opportunity to explain how a dog whistle works, too.
The concept of there being light that we can't see and sound that we can't hear is a really amazing one when one hears of it first. I certainly was intrigued by it when I learned about this as a child.
1. This may not be so easy and may not be desirable, see Cleonis' comment below