Can we find an example where $U$ is not connected and the partition of $p^{-1}(U)$ into slices is not unique? This is an extension of the following question from Munkres. Section 53-problem 2.
Let $p: E \to B$ be continuous and surjective. Suppose that $U$ is an open set of $B$
that is evenly covered by $p$. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ into slices is unique.
I could solve this problem but one question that I could not answer: 
Can we find an example where $U$ is not connected and the partition of $p^{-1}(U)$ into slices is not unique? 
I tried simple examples of the covering map $f : \mathbb{R} \to S^1$ and the non-covering maps $g : \mathbb{R}^{+} \to S^1$ without any success.

Simplest example:
Take $B = \{a,b\}$ and $E = B \times \{0,1\}$ with $p$ being the map that forgets the second coordinate. Now let $U = B$. We could take as the two slices, either $$U_1 = \{(a,0), (b,0)\} \mbox{ and } U_2 = \{(a,1), (b,1)\}$$ 
or
$$U'_1 = \{(a,0) , (b,1) \}\mbox{ and } U'_2 = \{(a,1), (b,0)\}.$$ 
