  In this paper, we introduce and develop the notion of a Manin triple for a
Lie superalgebra $\mathfrak g$ defined over a field of characteristic $p=2$. We
find cohomological necessary conditions for the pair $(\mathfrak g, \mathfrak
g^*)$ to form a Manin triple. We introduce the concept of Lie bi-superalgebras
for $p=2$ and establish a link between Manin triples and Lie bi-superalgebras.
In particular, we study Manin triples defined by a classical $r$-matrix with an
extra condition (called an admissible classical $r$-matrix). A particular case
is examined where $\mathfrak g$ has an even invariant non-degenerate bilinear
form. In this case, admissible $r$-matrices can be obtained inductively through
the process of double extensions. In addition, we introduce the notion of
double extensions of Manin triples, and show how to get a new Manin triple from
an existing one.
