This may well be the class X2 filter capacitors on the analogue board. These explode and produce smoke quite easily when they're old, but 1) they're not an essential part of the power supply, so the unit will continue to function perfectly in that condition, and 2) it's perfectly safe to do so.
I've had loads of BBC Model Bs, a couple of Mac Pluses, and Apple IIe, and some others had this happen to them. The parts probably cost something like $2 in the US, and anybody who can fix TVs could probably do the repair for beer money.