A family of Dirac-Born-Infeld (DBI) Lagrangians has been known and widely considered in many different fields of physics. In the last several decades they find a significant room of application in cosmology and inflation theory. Dynamics governed by DBI Lagrangians are closely related to dynamics of the tachyon fields. We are focussed on quantum dynamics of tachyon-like systems with a special attention on nonarchimedean spaces, their p-adic and adelic aspects and connection with dynamics on ``real-Archimedean`` space.