We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $Δ(G)\leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar [{\em Trans. Amer. Math. Soc., 2007}] on the size of planar graphs with positive combinatorial curvature.