In this work, we set up a theory of $p$-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of $k$-th invariant differentials over the Shimura curves we are interested in, for any $p$-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any $p$-adic weight. Moreover, our sheaves can be put in $p$-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the $\mathrm{U}$ operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.