  The present paper is devoted to studying local derivations on the Lie algebra
$W(2,2)$ which has some outer derivations. Using some linear algebra methods in
\cite{CZZ} and a key construction for $W(2,2)$ we prove that every local
derivation on $W(2, 2)$ is a derivation. As an application, we determine all
local derivations on the deformed $\mathfrak{bms}_3$ algebra.
