And the answer is . . . . . .
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"I'm very busy."
Before posting the result, I thought I'd better google "drinking lizard," just to make doubly sure of my facts. I was quite surprised to find that some sites translate this saying as meaning, "I'm in a hurry." I suspect these sites may be non-Oz and are being misled by "flat out," which certainly does mean "in a hurry" or "moving quickly." However, in Australia, when the words "flat out" are used in conjunction with any reference to imbibing reptiles, no matter how remote, it only ever means one thing - "I'm very busy." Certainly, that's what it means in my office! For example, imagine I'm up to pussy's bow in overdue paper work and a colleague drops by my desk for a chat. It would be entirely appropriate for me to say, "I'm flat out like a lizard drinking right now, mate. I'll catch ya later!"
Congratulations to everyone who got it right and thank you to everyone for participating.