I believe the fallowing is a folklore:
Suppose $f:\mathbb{H}^n \rightarrow \mathbb{H}^m$ (or actually between open subsets of them) is of class $\mathcal{C}^1$ (can we assume to be only Frechet differentiable?) in the sense that the Frechet derivative $d_qf \in \mathcal{M}_{n,m}(\mathbb{H})$ defined by $$ lim_{h \rightarrow 0}\frac{||f(q+h)-f(q)-d_qf(h)||}{||h||}=0$$ exist in every point and is continuous as a function $df:\mathbb{H}^n \rightarrow \mathcal{M}_{n,m}(\mathbb{H})$. Then $f(q)=Aq + q_0$ for some $A \in \mathcal{M}_{n,m}(\mathbb{H})$ and $q_0 \in \mathbb{H}$.
The only reference I know for the case $n=1$ is by A. Sudbery. There a different definition of "differentiability" is used although from Frechet differentiability the former fallows (as well as its right version). The bigger problem is that reasoning involves quaternion differentials as well as passing to some formal complex derivatives. Does one know other references (possibly to the general situation) or maybe a simpler reasoning?