Non-trivial bin-packing instance with 5 objects.

No, there is no such instance for any $n\le 5$. Indeed, assume to the contrary. Pick the smallest $n$ for which $O<A$, where $O$ is the number of bins in an optimal packing and $A$ be the number of bins in a packing found by best-fit-decreasing heuristic. Enumerate the objects in such an order that $s_1\ge s_2\ge\dots\ge s_n$. Now we use the following simple observations.

If $O=1$ then $A=1$.

Assume that in an optimal packing there exists a bin containing only one object. But, since $s_1\le v$, without loss of generality we may assume that the bin contains Object 1 and the other objects placed into the first bin by best-fit-decreasing heuristic. Removing these objects from the object list, we reduce the problem to a smaller number of objects, which contradicts the minimality of $n$.

The above observations imply that $O=2$, because otherwise since $n<6$ in each optimal packing there exists a bin containing only one object. Now fix and optimal packing and consider a bin containing Object $1$. If the bin contains at lest two other objects then it can be easily checked that $A=2$. If the bin contains at most one other object $i$ then the other bin contains all remaining objects. Then in the packing created by best-fit-decreasing heuristic the first bin contains at least Object $1$ and Object $j$ with $j\le i$, so the second bin contains all other objects, because it is able to contain them, so $A=2$.