Unboundedness of a $p$-adic Gaussian distribution

The intensive development of mathematical physics over non-Archimedean number fields has led to the emergence of many new mathematical constructions. In particular, a $p$-adic Gaussian distribution was introduced that lies at the basis of $p$-adic quantum mechanics with $p$-adic-valued functions. In this paper it is proved that, in contrast to the real theory, a Gaussian distribution in the $p$-adic case is not a measure, and the corresponding linear functional is unbounded on the space of continuous functions.