The stability of planar films of liquid 4 He at Tϭ0 K without a supporting potential is investigated. For this purpose, the third-sound velocity, c 3 , is examined within the framework of two different theoretical descriptions: ͑i͒ the correlated-basis functions theory in conjunction with the hypernetted-chain approximation and ͑ii͒ the nonlocal density functional theory. All the calculations yield negative values of c 3 2 . In particular, the behavior of the chemical potential as a function of the coverage provides convincing evidence in favor of the instability of all the analyzed free films. Furthermore, the analysis of the trend of thick films leads to the plausible conjecture that free planar films with finite coverage would be always unstable. The conclusion of the present work matches well with the nonwetting phenomenon of Rb and Cs substrates by a liquid 4 He film at Tϭ0 K and with the hydrodynamic prediction that a free semi-infinite system with a flat surface is unstable.