In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with $σ_4(G)\geq n-1$ contains a spanning tree with at most $3$ leaves. In this paper, we prove an analogue of Kyaw's result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with $σ_5(G)\geq n-1$ contains a spanning tree with at most $4$ leaves. Moreover, the degree sum condition `$σ_5(G)\geq n-1$' is best possible.