The celebrated Berezinskii-Kosterlitz-Thouless (BKT) phase transition refers to a topological transition characterized, e.g., by the dissociation of vortex-antivortex pairs in two-dimensional (2D) systems. Such unusual phase has been reported in various types of materials, but never in the new class of systems made by one-unit-cell-thick (1UC) ferroelectrics (also coined as 2D ferroelectrics). Here, the use of a first-principles-based effective Hamiltonian method leads to the discovery of many fingerprints of a BKT phase existing in-between the ferroelectric and paraelectric states of 1UC tin tellurium being fully relaxed. Moreover, epitaxial strain is found to have dramatic consequences on the temperature range of such BKT phase for the 1UC SnTe. Consequently, our predictions extend the playground of BKT theory to a novel class of functional materials, and demonstrate that strain is an effective tool to alter BKT characteristics there.