Is there a good notion of holonomic $D$-modules on rigid analytic spaces?
Yes. Although it is only beginning to be developed.
You probably want to start with Berthelot: D-modules arithmétiques I : Opérateurs différentiels de niveau fini and Introduction à la théorie arithmétique des D-modules, and other papers that can be found at http://perso.univ-rennes1.fr/pierre.berthelot/ Section 5 of the second paper I mentioned is perhaps most relevant.
There is also a recent paper of Caro which I cannot find online called 'Holonomie sans structure de Frobenius et criteres d'Holonomie' which removes the necessity of the Frobenius action from Berhelot's work. I suppose he would send you a copy of upon request.
Finally, in a piece of shameless self-advertising, Konstantin Ardakov and I recently put a preprint on the arXiv http://arxiv.org/abs/1102.2606 part of which seeks to find a framework to further develop the theory.
Update: Caro's paper mentioned above now seems to be available here: http://aif.cedram.org/item?id=AIF_2011__61_4_1437_0 although you need a subscription to access it.
Further update (10th Feb 2015): Apologies for the further self-advertising but Konstantin Ardakov and I now have two further preprints on the topic of D-modules on rigid analytic spaces http://arxiv.org/abs/1501.02215 and http://arxiv.org/abs/1502.01273. There is no mention of holonomicity in either of these but there seems to be a natural definition of the notion in the framework outlined in these. Whether this definition behaves as one might hope is likely to be discussed in future work.