  Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed
and $\mathcal O$ unramified, and let $\Lambda$ be an $\mathcal O$-order in a
separable $K$-algebra. We call a $\Lambda$-lattice $L$ rigid if ${\rm
Ext}^1_{\Lambda}(L,L)=0$, in analogy with the definition of rigid modules over
a finite-dimensional algebra. By partitioning the $\Lambda$-lattices of a given
dimension into "varieties of lattices", we show that there are only finitely
many rigid $\Lambda$-lattices $L$ of any given dimension. As a consequence we
show that if the first Hochschild cohomology of $\Lambda$ vanishes, then the
Picard group and the outer automorphism group of $\Lambda$ are finite. In
particular the Picard groups of blocks of finite groups defined over $\mathcal
O$ are always finite.
