Strictly speaking, there is no such thing as a perfectly-competitive market in the real world. The formal definition of perfect competition requires an infinite number of firms with perfect substitution and zero market power. That doesn't happen in real life.
However, there are some industries that get fairly close to perfect competition, where there are a very large number of firms, with very similar products, who operate at very small profit margins.
A good example is farmers. For most agricultural products, ranging from corn to soybeans to tomatoes, there are a large number of individual farms, each of which produces a very small portion of the global market. Each farmer has very little say over what price they can charge for their harvest; there is a market price they have to meet, and if they don't, no one will buy. As a result, farmers have very low profit margins, and it is not uncommon for farmers to have a bad year or two and be driven to bankruptcy.
Another example is gold. While gold itself is very expensive, gold mining is actually not a very profitable business, because there are so many different companies selling gold, all trying to undercut each other's prices. Gold markets are also so closely linked to financial markets that we can watch the global price of gold rise and fall minute-by-minute.
But perhaps the closest we ever actually get to perfect competition in the real world is currency. Currency exchange markets are constantly trading back and forth---dollars for Euros, Euros for yen, yen for yuan, yuan for dollars. Billions of dollars change hands every second. Theoretically it would be possible to make a profit by buying currencies when they are cheap and selling them when they are expensive (speculation) or by exploiting inconsistencies in the prices of different currencies (arbitrage).
Theoretically, it could turn out that $1 buys 0.8 Euros or 7 yuan, but 0.8 Euros buys only 6 yuan, so you could buy yuan with dollars, then buy Euros with yuan, then buy dollars with Euros and make a profit. But in practice, competition in the currency market is so fierce---and so fast---that these kinds of inconsistencies disappear in milliseconds.
Profiting from arbitrage in currency exchange is only possible if you are one of the few companies that has access to high-frequency-trading systems that can actually make trades in that short a time. Thus, currency exchange is almost perfect competition: Prices are the same for everyone, nobody has power over the price, and profits are zero.