The longest common extension (LCE) problem is to preprocess a given string ω of length n so that the length of the longest common prefix between suffixes of ω that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z2 + n/t ) words of space which answers LCE queries in O(1) time and can be built in O(n log δ) time, where 1 ≤ T ≤ √n is a parameter, z is the size of the Lempel-Ziv 77 factorization of ω and φ is the alphabet size. The proposed LCE data structure does not access the input string ω when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following: For highly repetitive strings where the z2 term is dominated by n/x, we obtain a constant-time and sub-linear space LCE query data structure. Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable and for φ ≤ 2o(log n). The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.