Is it possible for an iPad - or any iOS product - to get a virus? And if so, have there been any recorded instances of this?

(I'm excluding the so-called "jail broken" devices.)
Yes, security vulnerabilities that would allow people to hack or infect Macs and iOS products do exist, though they're not exploited very often.

Pwn2Own is an annual contest where competitors try to hack into machines to win them as prizes. Both Macs and iOS products are hacked most years.

Here is one more story about an exploit that was patched:


  Friday, Apple released an update to its mobile operating system (iOS 4.3.4) which patches a couple of vulnerabilities that left a door open for malware infections on the iPad, 3rd and 4th generation iPod touch, iPhone 4, and iPhone 3GS.
  
  Apple's update describes the CoreGraphics vulnerability as "A buffer overflow…in FreeType's handling of TrueType fonts. Viewing a maliciously crafted PDF file may lead to an unexpected application termination or arbitrary code execution."
  
  [...]
  
  This update comes on the heels of an alert from German IT group BSI, which warned of an unpatched vulnerability that would "allow attackers to gain access to the entire system with administrative privileges."


Apple does "verify" any software for their App Stores, but developers have been known to sneak in features without Apple's testers seeing them, so don't expect this to entirely protect you.