A celebrated theorem of Chris Miller states that if R is an o-minimal expansion of the field of real numbers then either R is polynomially bounded or the exponential function is definable in R. After introducing an analogue of o-minimality for expansions of algebraically closed valued fields (called C- minimality), the aim of the talk is to show that every C-minimal expansion of a valued field (K, v) having value group Q is polynomially bounded. In particular, we obtain that any C-minimal expansion of valued fields like $C_p$, $F^{alg} ((t^Q ))$ are polynomially bounded. This is a joint work with Françoise Delon.