  We show that the Heisenberg Lie algebras over a field $\mathbb{F}$ of
characteristic $p>0$ admit a family of restricted Lie algebras, and we classify
all such non-isomorphic restricted Lie algebra structures. We use the ordinary
1- and 2-cohomology spaces with trivial coefficients to compute the restricted
1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We
describe the restricted 1-dimensional central extensions, including explicit
formulas for the Lie brackets and $\cdot^{[p]}$-operators.
