We show that the Bogomol’nyi-Prasad-Sommerfield (BPS) property is a generic feature of all models in ( 1 + 1 ) dimensions that does not put any restriction on the action. Here, by BPS solutions we understand static solutions that (i) obey a lower-order Bogomolny-type equation in addition to the Euler-Lagrange equation, (ii) have an energy that only depends on a topological charge and the global properties of the fields, but not on the local behavior (coordinate dependence) of the solution, and (iii) have zero pressure density. Concretely, to accomplish this program we study the existence of BPS solutions in field theories where the action functional (or energy functional) depends on higher than first derivatives of the fields. We find that the existence of BPS solutions is a rather generic property of these higher-derivative scalar field theories. Hence, the BPS property in 1 + 1 dimensions can be extended not only to an arbitrary number of scalar fields and k-deformed models, but also to any (well-behaved) higher-derivative theory. We also investigate the possibility to destroy the BPS property by adding an impurity that breaks the translational symmetry. Further, we find that there is a particular impurity-field coupling that still preserves one-half of the BPS-ness. An example of such a BPS kink-impurity bound state is provided.