This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96 % compared to the algorithm from Gorodetsky et al. (Comput. Methods Appl. Mech. Eng. 347, 59-84 2019).