The probability that a student is not a swimmer is $\frac{1}{5}$. The probability that out of 5 students selected at random 4 are swimmers is

The probability that a student is not a swimmer is $P(S') = \frac{1}{5}$, so the probability a student is a swimmer is $P(S) = 1 - \frac{1}{5} = \frac{4}{5}$.

We seek the probability that exactly 4 out of 5 randomly selected students are swimmers, a binomial scenario with $n=5$, $k=4$, $p = \frac{4}{5}$, and $q = \frac{1}{5}$.

The binomial probability formula gives
$P(X=4) = C(5, 4) \cdot \left(\frac{4}{5}\right)^4 \cdot \left(\frac{1}{5}\right)^1$.

Since $C(5,4) = 5$, this simplifies to
$5 \cdot \frac{4^4}{5^4} \cdot \frac{1}{5} = \frac{4^4}{5^4} = \left(\frac{4}{5}\right)^4$.

This matches option A, so the answer is $\boxed{\text{A}}$.