Starting from p-adic string theory with tachyons, we introduce a new kind of non-tachyonic matter which may play an important role in evolution of the Universe. This matter retains nonlocal and nonlinear p-adic string dynamics, but does not suffer of negative square mass. In space-time dimensions D = 2 + 4k, what includes D = 6, 10, ..., 26, the kinetic energy term also maintains correct sign. In these spaces this p-adic matter provides negative cosmological constant and time-dependent scalar field solution with negative potential. Their possible cosmological role is discussed. We have also connected non-locality with string world-sheet in effective Lagrangian for p-adic string.