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We propose a model of a quantum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">N</mi></semantics></math></inline-formula>-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of <i>p</i>-adic numbers. As in the standard complex setting, states and observables of a <i>p</i>-adic quantum system are implemented by suitable linear operators in a <i>p</i>-adic Hilbert space. In particular, owing to the distinguishing features of <i>p</i>-adic probability theory, the states of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">N</mi></semantics></math></inline-formula>-dimensional <i>p</i>-adic quantum system are implemented by <i>p</i>-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable <i>p</i>-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a <i>p</i>-adic quantum system. Eventually, we focus on the special case where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">N</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, thus providing a description of <i>p</i>-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed.