Nodal chains in which two nodal rings connect at one point were recently discovered in non-symmorphic electronic systems and then generalized to symmorphic phononic systems. In this work, we identify a new class of planar nodal chains in non-symmorphic phononic systems, where the connecting rings lie in the same plane. The constituting nodal rings are protected by mirror symmetry, and their intersection is guaranteed by the combination of time-reversal and non-symmorphic twofold screw symmetry. The connecting points are fourfold degenerate while those in previous works are twofold degenerate. We found 8 out of 230 space groups that can host the proposed planar nodal chain phonons. Taking wurtzite GaN (space group No. 186) as an example, the planar nodal chain is confirmed by first-principles calculations. The planar nodal chains result in two distinct classes of drumhead surface states on the [10(–1)0] and the [0001] surface Brillouin zones. Our finding reveals a class of planar nodal chains in non-symmorphic phononic systems, expanding the catalog of topological nodal chains and enriching the family of topological surface states.