Definable measure preserving isomorphisms of $p$-adic semialgebraic sets

Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a definable (hence semialgebraic) subset $Z$ of $K^n$ of $p$-adic dimension $d$ and let $C$ be a definable subset of $Z$ of strictly lower dimension.

My question is:

Is there a definable isomorphism $f: Z \rightarrow Z \setminus C$ such that $\mid {\rm Jac} f \mid=1$?

I know the answer for $n=1$ is no but I need to know whether it is still no for $n>1$.

Thank you

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For sake of completeness: also posted on MathOverflow. –  Asaf Karagila Oct 3 '11 at 20:42
yes, that is right. –  user17090 Oct 3 '11 at 20:46
You should probably post a link here on the MO question as well. –  Asaf Karagila Oct 3 '11 at 20:47
I don't know how to do that. –  user17090 Oct 4 '11 at 8:52