Consider for instance $A := \mathbb{Z} \cup \{ a, b\}$, where we order the elements from $\mathbb{Z}$ in the usual way, and we define $a < b$, thus creating a partial order on $A$. Because there is no $x \in A$ such that $x < a$, $a$ is a minimal element, but not the smallest element because $a < x$ is not satisfied for all $x \in A$. After all, $a < x$ is not defined for any $x \in \mathbb{Z}$.
Clearly there is no minimal element in $\mathbb{Z}$, so the condition is satisfied.
One doesn't have to use $\mathbb{Z}$ in particular, but what made it easy is the fact that it is an infinite set (in the sense that I couldn't think of any counterexamples for finite sets). The reason why I used $\mathbb{Z}$ is just because it comes with a default order according to which it has no minimal element (or smallest, for that matter).