Summary: \(p\)-Adic quantum mechanics is constructed from the Dirac-von Neumann axioms identifying quantum states with square-integrable functions on the \(N\)-dimensional \(p\)-adic space, \(\mathbb{Q}_p^N\). This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The \(p\)-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a \(p\)-adic heat equation by a temporal Wick rotation. This \(p\)-adic heat equation describes a particle performing a random motion in \(\mathbb{Q}_p^N\). The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as \(p\)-adic de Broglie waves. These mathematical waves have all wavelength \(p^{-1}\). In the \(p\)-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the \(p\)-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by ‘quantum waves’. In the \(p\)-adic framework, in the double-slit experiment, each particle goes through one slit only. The \(p\)-adic quantum mechanics is an analog (or model) of the standard one under the hypothesis of the existence of a Planck length. The precise connection between these two theories is an open problem. Finally, we propose the conjecture that the classical de Broglie wave-particle duality is a manifestation of the discreteness of space-time.