Many people wash their fruits because they say that supermarkets put a chemical spray on them to keep them nice for a long time without rotting. (They are concerned that eating food with that spray is unhealthy for our body.)
Is this true?
Your question asks about "supermarkets", but I have interpreted that as synecdoche - that you really mean to ask about the whole fruit supply chain, and not just the retailers at the end of it.
Yes, the agricultural industry uses sprayed chemicals to prolong the shelf-life of food.
I list two examples below - but there may be many more.
I haven't listed pesticides that may still be on the fruit, on the grounds that is generally to increase yield, not extend shelf-life.
Both of these waxes are “food-grade waxes” and have been used on food for decades. The FDA has labeled both of these waxes safe for human consumption. Both of these natural waxes are complex mixtures and contain some of the same components found in the wax of an apple.
The FDA recommend washing fruit. We have a related question about the best way to do this: Do fruit/vegetable washes effectively remove pesticides?
The reason given is to remove dirt, but especially microbes. Even if you doubt the existence of sprayed chemicals, or are confident that the FDA has correctly assessed that trace amounts are not a danger, you should wash your fruits and vegetables anyway.