No, saving it saves the animation (all the frames of the file). However, when you look at the thumbnail to open it it only shows the first frame of the animation. If you open it in any program that is not an animation program THEN it loses the animation effect and only saves the first frame. Even if you open it in an animation program it shows all the consecutive file frames rather than as functioning anmation.
What you have to do is save it to somewhere on your computer and leave it alone. It will be saved as an animation. Then when you want to send it in an email or post it on facebook or whatever when prompted for the file name just chose the file name of the animation. If sending by email attach it as a photo or embed it directly in the email. Since I don't use facebook I don't know how they prompt you to add photos or animations, but there must be some point where they ask you the file name.
So, just right click on the animation, chose "save as" and either keep the file name or change it to one you like better and save it to your desktop or in some folder of your choice and leave it alone - don't try to open it any program that isn't an animation program. If you want to see it animate on your computer once you've saved it you'd have to send it as an attachment or embeded in an email sent to yourself. Once it's saved to your computer you can then send it out by email or put it on facebook or whatever. Just don't try to OPEN it in any program on your computer that isn't an animation program. If you try to open it in say Paint or Photoshop it will only show the image of the first frame of the animation and all the other frames will be lost.
I have a ton of animations saved on my computer and though in thumbnails they look like a single image they're still animations and will stay that way to email out or put on the web or whatever as long as I don't try to OPEN them except in my animation program. I've made that mistake before when learning to make animations. You just have to trust that it saved as an animation when saving to your computer and don't ever try to OPEN it on your computer unless you open it in an animation program.