  In this paper, we introduce poly-Bernoulli numbers with level $2$, related to
the Stirling numbers of the second kind with level $2$, and study several
properties of poly-Bernoulli numbers with level $2$ from their expressions,
relations, and congruences. Poly-Bernoulli numbers with level $2$ have strong
connections with poly-Cauchy numbers with level $2$. In a special case, we can
determine the denominators of Bernoulli numbers with level $2$ by showing a von
Staudt-Clausen like theorem.
