Profit is not the same thing as economic rent. You can think of economic rent as the portion of revenues deriving purely from the scarcity of some resource you hold, e.g. land in Manhattan (but not the stuff involved in administering or maintaining it), or the rights to a patent, or skills or credentials that are underrepresented in a market thanks to cultural factors. That has almost nothing to do with profit; although you can profit from economic rents, you can profit from other things too, and almost all transactions involve factors other than rents. A perfect market would eliminate rents, but it would not eliminate profits -- though it would drive down profits to the minimum necessary to motivate transactions.

There are rents involved in the production of most material goods, because natural resources are almost always scarce in this sense. However, as a counterexample we could imagine e.g. a frontier situation where land was essentially free and the prices of agricultural goods directly reflected the costs and effort involved in cultivating them.