There is quite a tradition in the study of free objects in functional analysis. In particular, free Banach lattices have found quite a number of interesting applications transferring properties of Banach spaces to properties of Banach lattices. In this talk, we address the existence of such objects in the realm of p-Banach spaces, where $0<p<1$, and we apply their properties to produce some examples of projective p-Banach lattices. This is an ongoing work with Pedro Tradacete (ICMAT) and Nazaret Trejo (UCM).