$p$-Adic string theory was introduced in 1987 by construction of an analogue of the Veneziano scattering amplitude replacing real world-sheet by the $p$-adic one. It was shown that product of ordinary crossing symmetric Veneziano amplitude and its $p$-adic counterparts over all primes $p$ is a constant. This convergent infinite product of amplitudes is an example of connection between ordinary and $p$-adic strings. Investigation of $p$-adic strings was improved by invention of an effective Lagrangian. This Lagrangian with scalar field describes not only four-point scattering amplitude but also all higher ones at the tree level. This line of research has led to new insights on the role of $p$-adic strings in string theory. However, $p$-adic strings have been mainly treated as an auxiliary tool to better understand ordinary strings. However, if ordinary matter has its origin in ordinary strings, then there should be $p$-adic matter related to $p$-adic strings. In this talk, I will start with a brief review of some basic properties of $p$-adic strings and their connections with ordinary strings. Then I will consider a slight modification of the Lagrangian of $p$-adic open string to get a non-tachyonic $p$-adic matter. It will be shown that one obtains a new well defined scalar $p$-adic field. Equation of motion for this field is nonlocal and nonlinear. Some cosmological consequences are investigated in a weak field approximation. It will be presented cosmological solution with an exponential expansion (and contraction) of a closed universe.