I don't think so unless it has changed in the past 10 years or so, a couple of my mates did their driving tests there because their dad was stationed in Gibraltar but they definitely weren't citizens. They said it was very easy although I suppose that may have changed and you should hope you don't have to cross the runway on your test as that can occasionally be closed for quite a long time I understand. I've also got no idea how long the waiting list is for tests - that could potentially screw your plans up completely.
I probably wouldn't recommend going there as being worth the effort, maybe just do an intensive course in the UK and see if you can pass that way.
//edit ah just seen elmuzzy66.'s post, I hadn't realised there was a residency requirement although I'm surprised my mates fulfilled it if I'm honest given they'd been at university for a year or two at that point.