Step 1: Idea:
Match the unit of $k$ with the order.
Step 2: Result:
Only a first order rate law $r = k[A]$ gives $k = r/[A]$ with units of $\text{mol L}^{-1}\text{s}^{-1} / \text{mol L}^{-1} = \text{s}^{-1}$.
Final Answer:
The reaction is of first order, option (B).
\[ \boxed{\text{First order}} \]