I have upgraded my own iPad 3, my Wife's iPad 2 and my new iPod Touch, all with 64 GBs of storage space, and every device had a successful upgrade to iOS 9 with no issues.
If your iPad was near full (2 GBs of free storage space or less) this is but only one reason why your older iPad isn't coping well with iOS 9.
You need to backup all of your data on your iPad, either to iTunes on a computer, through Apple's own iCloud OR a dedicated Wireless WiFi hard drive, and then start deleting items off of your iPad.
You should start with paring down any music, movies and photos you may have on your iPad.
Then look at apps that you may no longer use and delete these (you can always redownload deleted apps at a later time.
Ideally, your iDevice should maintain a minimum free storage amount of 4- 6 GBs or greater on any iDevice, plus, more than 50% battery status to ensure proper operation, as well as, to ensure the possibility of a successful iOS upgrade/update installation.
In addition, it is best to upgrade an iDevice connected up to computer running iTunes with you iDevice sync'd and backed up through iTunes rather than Over The Air (OTA) using a wireless WiFi connection.