Go to the field's Properties, and under Options tick the box that says "Allow user to enter custom text".
Thanks for the reply. I went ahead and tried that, and had a couple of issues.
1. Allowing users to enter custom text stops the drop down box from working. If I type in 'Vie' for Vietnam, the drop down box doesn't bring anything up.
2. If I type in the correct country name "Vietnam", the corresponding code in the adjacent box does not appear and instead the word 'Vietnam' appears where normally a 3 digit number would had I selected the country name from the drop down box.
I see what you mean now... In that case, I'm afraid it's not possible. The closest you can get to it is to use a separate field (a text field) as the "filter" for the drop-down. You can set that up using this script I've developed: Custom-made Adobe Scripts: Acrobat -- Create Live Filtered List