We consider the classical null $p$-brane dynamics in $D$-dimensional curved backgrounds and apply the Batalin–Fradkin–Vilkovisky approach for BRST quantization of general gauge theories. Then we develop a method for solving the tensionless $p$-brane equations of motion and constraints. This is possible whenever there exists at least one Killing vector for the background metric. It is shown that the same method can be also applied for the tensile $1$-branes. Finally, we give two examples of explicit exact solutions in four dimensions.