This paper describes a time delay estimation (TDE) technique using compressive sensing (CS) off the grid, which estimates the channel impulse response in a continuous time domain. The TDE can be formulated into a sparse signal reconstruction problem where the CS technique can be applied. Previous works have used standard finite dimensional CS with evenly discretized grids. However, the actual time delays will not always lie on the discrete grid, and this mismatch between the actual and discretized time delays results in reconstruction degradation. To overcome the basis mismatch, a TDE technique using an off the grid CS framework is proposed by modifying the scheme in the off the grid direction of arrival (DOA) estimation [Xenaki and Gerstoft, J. Acoust. Soc. Am. 137(4), 1923-1935 (2015)]. The effectiveness of the suggested method is demonstrated on real data from a water tank experiment. The off the grid CS TDE is shown to have super-resolution, which enables close arrivals to be distinguished.