There are certain foods which hamsters cannot eat, though peaches are not one of them. You will be able to give your hamster many different fruits and vegetables and they will not affect them adversely whatsoever.
Some of the different fruits which hamsters can eat with no problems at all include basil, blueberries, blackberries, cabbage (only in small amounts), carrots, broccoli, and much more. You will find that chestnuts, celery, and cherries are also acceptable to feed most hamsters.
If you do feed your hamster peaches though, you should make certain that they are small slices and that they only get them every once in a while.