A classical topic in the theory of Banach structures is the automatical continuity of derivations. From 1968, when Johnson and Sinclair proved the continuity of derivations acting on semisimple associative Banach algebras, until now, several algebraic conditions on a Banach algebra \(A\) which ensure the continuity of its derivations have been established. A relevant result has been proved recently by A. Villena: derivations on semisimple Banach-Jordan algebras are continuous. In this work, the authors generalize this last fact for derivations acting on Banach-Jordan pairs, which are a natural extension of Banach-Jordan algebras. Moreover, Banach-Jordan pairs are important in themselves by their connection with the geometry of bounded symmetric domains. A Banach-Jordan pair is a pair \((V_+,V_-)\) of Banach spaces endowed with two continuous quadratic mappings \(Q_\sigma:V_\sigma\to \text{ Hom}(V_{-\sigma}\to V_\sigma)\) (\(\sigma\in\{+,-\}\)) satisfying certain algebraic conditions. A pair \(D=(D_+,D_-)\in\text{ End}(V_+)\times\text{ End}(V_-)\) is called a \textsl{derivation} if the condition \[ D_\sigma(\{x,y,z\})= \{D_\sigma(x),y,z\}+\{x,D_{-\sigma}(y),z\}+\{x,y,D_\sigma(z)\}, \] holds for any \(x,z\in V_\sigma\), \(y\in V_{-\sigma}\), \(\sigma\in\{+-\}\), where \[ \{x,y,z\}=(Q_\sigma(x+z)-Q_\sigma(x)-Q_\sigma(z))(y). \] The main result of this paper is the continuity of derivations on semiprimitive Jordan-Banach pairs. The proof uses well-known ideas, such as the separating ideal, and other recent techniques from purely algebraic Jordan theory. This makes its reading very difficult for the non-expert on the subject. This work can also be considered as a nontrivial extension of the automatic continuity of derivations acting on JB\(^*\)-triples.