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For a $p$-adic curve $X$, we study conditions under which all classes in the$n$-torsion of $Br(X)$ are $\mathbb{Z}/n$-cyclic. We show that in general notall classes are $\mathbb{Z}/n$-cyclic classes. On the other hand, if $X$ hasgood reduction and $n$ is prime to $p$, of if $X$ is an elliptic curve over$\mathbb{Q}_p$ with split multiplicative reduction and $n$ is a power of $p$,then we prove that all order $n$ elements of $Br(X)$ are $\mathbb{Z}/n$-cyclic.Finally, if $X$ has good reduction and its function field $K(X)$ contains all$p^2$-th roots of $1$, we show the existence of indecomposable divisionalgebras over $K(X)$ with period $p^2$ and index $p^3$.

Cyclicity and indecomposability in the Brauer group of a $p$-adic curve