  Let $\mathbf{k}$ be an algebraically closed field of arbitrary
characteristic, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra and
let $V$ be a $\Lambda$-module with stable endomorphism ring isomorphic to
$\mathbf{k}$. If $\Lambda$ is self-injective, then $V$ has a universal
deformation ring $R(\Lambda,V)$, which is a complete local commutative
Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if
$\Lambda$ is further a Frobenius $\mathbf{k}$-algebra, then $R(\Lambda,V)$ is
stable under syzygies. We use these facts to determine the universal
deformation rings of string $\Lambda_{m,N}$-modules whose corresponding stable
endomorphism ring is isomorphic to $\mathbf{k}$, and which lie either in a
connected component of the stable Auslander-Reiten quiver of $\Lambda_{m,N}$
containing a module with endomorphism ring isomorphic to $\mathbf{k}$ or in a
periodic component containing only string $\Lambda_{m,N}$-modules, where $m\geq
3$ and $N\geq 1$ are integers, and $\Lambda_{m,N}$ is a self-injective special
biserial $\mathbf{k}$-algebra.
