Is independence preserved if a random entry in an independent sequence is replaced by a constant?

Let $$\xi=(\xi_1,\ldots,\xi_n)$$ be a sequence of independent random variables. Let us pick an index $$\nu\in \{1,\ldots,n\}$$, and replace the entry $$\xi_\nu$$ by a constant $$c$$. The rest of the $$\xi_i$$ remain unchanged.

Question: Is it true that the altered sequence remains independent, even if $$\nu$$ is random, possibly dependent on $$\xi$$?

Note: If $$\nu$$ is a fixed constant index, then the independence is clearly preserved. It is less clear, however, what happens if $$\nu$$ is random and dependent on $$\xi$$. For example, we may pick the first hit to some set, or apply any other data-dependent rule to pick the index $$\nu$$.

The answer is no. E.g., assume that $$n=2$$, $$\xi_i=X_i$$, and $$X_1,X_2,\nu$$ are any random variables (r.v's) each with values in the set $$\{1,2\}$$ and with $$P(\nu=1)=p\in(0,1)$$. With these conditions in place, the dependence between $$X_1,X_2,\nu$$ may be arbitrary. For instance, we may suppose that $$\nu$$ is independent of $$X_1,X_2$$, or we may suppose that $$\nu=X_1$$, or ... .
Take any $$c\notin\{1,2\}$$ and let $$Y_1,Y_2$$ be obtained from $$X_1,X_2$$ by replacing $$X_\nu$$ by $$c$$, so that
(Y_1,Y_2)=\left\{ \begin{aligned} (c,X_2) &\text{ if }\nu=1,\\ (X_1,c) &\text{ if }\nu=2. \end{aligned} \right. Then $$P(Y_1=c,Y_2=c)=0\ne p(1-p)=P(\nu=1)P(\nu=2)=P(Y_1=c)P(Y_2=c).$$ So, $$Y_1,Y_2$$ are not independent.