We define wedge-lifted codes, a variant of lifted codes, and we study their locality properties. We show that (taking the trace of) wedge-lifted codes yields binary codes with the $t$-disjoint repair property ($t$-DRGP). When $t= N^{1/2d}$, where $N$ is the block length of the code and $d\geq 2$ is any integer, our codes give improved trade-offs between redundancy and locality among binary codes.