We outline a program aimed to set the foundations of a $p$-adic theory of quantum information. The approach we propose starts by first providing a suitable notion of a $p$-adic Hilbert space over a quadratic extension of the filed of $p$-adic numbers $\mathbb{Q}_p$. A state for a $p$-adic quantum system is defined as a functional acting on the (ultrametric) Banach $*$-algebra of $p$-adic bounded observables. The statistical interpretation of the theory is then completed by introducing the notion of a SOVM as a suitable $p$-adic counterpart to a POVM of the complex quantum theory. To extend the formalism to compound systems, and to introduce a proper notion of entanglement in $p$-adic quantum mechanics, we next address the construction of the tensor product of $p$-adic Hilbert spaces. To this end, we first consider two $p$-adic Hilbert spaces as simple vector spaces and define their algebraic tensor product. Then, we introduce a norm by adapting the projective norm used in the tensor product of complex Banach spaces to our case study. By metrically completing the resulting $p$-adic normed space, and after providing a suitable inner product together with an orthonormal basis, we obtain the tensor product of $p$-adic Hilbert spaces. Next, we show that this space is isomorphic to the space of trace-class operators. Finally, we study the tensor product of subspaces of $p$-adic Hilbert spaces, highlighting both the analogies and the non-trivial differences with respect to the standard complex case.