I installed Windows XP on VMware Player for software testing. Now I need the ability to save the image of this virtual partition, so I could restore it exactly as it was (and as fast as possible) many times later. Is this somehow possible in free version of VMware Player?
You probably want to clone your VMWare machine and store away the "clone image" for later reference. VMWare has a "Clone Virtual Machine Wizard" for this issue. Checkout the linked page for details.
But this is probably not possible using the free VMWare player, as the pages linked refer to the "VMWare Workstation". For a free solution, you might want to take a look at VirtualBox, which also describes cloning its disks. VirtualBox is available in a free version.
Edit: VirtualBox is able to handle VMWare's virtual machines -- so you can stick with your existing VM and simply replace the VMWare Player by VirtualBox.