  We construct a minimal representation of the orthosymplectic Lie supergroup
$OSp(p,q|2n)$, generalising the Schr\"odinger model of the minimal
representation of $O(p,q)$ to the super case. The underlying Lie algebra
representation is realized on functions on the minimal orbit inside the Jordan
superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction
is in line with the orbit philosophy. Its annihilator is given by a Joseph-like
ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a
natural generalization of a minimal representations to the context of Lie
superalgebras. We also calculate its Gelfand--Kirillov dimension and construct
a non-degenerate sesquilinear form for which the representation is
skew-symmetric and which is the analogue of an $L^2$-inner product in the
supercase.
