  We investigate possible cardinalities of maximal antichains in the poset of
copies $\langle \mathbb P(\mathbb X),\subset \rangle$ of a countable
ultrahomogeneous relational structure $\mathbb X$. It turns out that if the age
of $\mathbb X$ has the strong amalgamation property, then, defining a copy of
$\mathbb X$ to be large iff it has infinite intersection with each orbit of
$\mathbb X$, the structure $\mathbb X$ can be partitioned into countably many
large copies, there are almost disjoint families of large copies of size
continuum and, hence, there are (maximal) antichains of size continuum in the
poset $\mathbb P (\mathbb X)$. Finally, we show that the posets of copies of
all countable ultrahomogeneous partial orders contain maximal antichains of
cardinality continuum and determine which of them contain countable maximal
antichains. That holds, in particular, for the random (universal
ultrahomogeneous) poset.
