The role of mechanics is known to be of primary order in many arterial diseases; however determining mechanical properties of arteries remains a challenge. This paper discusses the identifiability of a Holzapfel-type material model for a mouse carotid artery, using an inverse method based on a finite element model and 3D digital image correlation measurements of the surface strain during an inflation/extension test. Layer-specific mean fiber angles are successfully determined using a five parameter constitutive model, demonstrating good robustness of the identification procedure. Importantly, we show that a model based on a single thick layer is unable to render the biaxial mechanical response of the artery tested here. On the contrary, difficulties related to the identification of a seven parameter constitutive model are evidenced; such a model leads to multiple solutions. Nevertheless, it is shown that an additional mechanical test, different in nature with the previous one, solves this problem