Problem with seppareating meshes

Hi!

I´ve got a big problem: I cannot sepparate meshes from their original face. I mean, I have several subdivisions in a face, and then I select one of them to make it an independent object clicking “P” on the keyboard, as all the tutorials says. But the subdivision is still there, stacked in the original face.I´ve been looking tutorial about it all morning, but still didn´t figure out what could be wrong. Could someone help me, please?

Cheers!

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But the subdivision is still there, stacked in the original face
What does this mean ? Show us.
Does the face actually separate to a new object but do you also see it as part of the original object. If so maybe your original object has duplicate faces. Check it. Select all vertices and remove any doubles (W / remove doubles)

Please show screenshots and attach a blend file to show exactly waht your problem is. Don’t rely on us having to guess