My chain is skipping only on the smallest sprocket, and it skips with every crank turn. The chain is not stretched. Also, chainrings are almost new. Could this freewheel be the reason, it doesn't seem worn out to me (although it's old):
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Sprocket, cassette and chain match together. An old set will work well but change one part and you'll be skipping. Often people will wear out their favorite gears, change the chain and then only the less favorite gears work reliably.
The problem is that the chain will actually ride up high on the teeth, eventually so high that they're not all in position and the chain-reaction of skip occurs; the sprocket teeth graze the chain links uncontrollably in that ever-so-painful sound.
The derailleur is surely not the problem (as you noticed). If it were you would first here the clack-clack-clack from the out of alignment.