I thought about possibility of occurring such event in the case of presence of fixed costs, but I would like to know others opinions.
Short answer: Yes, it is possible.
Decreasing average cost implies that marginal cost is less than average cost ($MC<AC$, which can be proved by simply taking the first derivative of $C(q)/q$). With constant marginal cost, there exists a simple linear cost function $C(q)=F+a\times q$ that satisfies the constant $MC$ condition, where the constant $F$ is the fixed cost and $a \times q$ is the variable cost, and the constant $a$ is $MC$. Therefore $AC=F/q+a$ is greater than $MC=a$.