Hi there

Try the following:

1. Confirm if any other internet functions work (I know you said that you have confirmed internet access, but test again anyhow)?  For example 5OD or YouTube.  If they dont work, its a network issue.  If they do work and iPlayer doesnt, continue.
2. Turn TV and router off completely (as in, unplug from the wall) leave off for a couple of minutes and turn all back on again.  Test iPlayer.  If it still doesnt work, continue..
3. If you can, remove iPlayer from the Application menu, and then refreshing internet content services.  Press the Home button, then goto Settings > Network > Refresh Internet Content.  If that still doesnt work, continue...
4. You may need to perform a factory reset.  I would suggest writing down all settings (ie picture/display and everything else) so you can input it later on.

Hope that helps.