This paper presents the first distributed algorithm to construct a spanner for arbitrary ad hoc networks under the physical signal-to-interference-and-noise-ratio (SINR) interference model. Spanner construction is one of the most important techniques for topology control in wireless networks, which intends to find a sparse topology in which only a small number of links need to be maintained, without substantially degrading the path connecting any pair of the nodes in the network. Due to the non-local property of interference, constructing a spanner is challenging under the SINR model, especially when a local distributed algorithm is desired. We meet this challenge by proposing an efficient randomized distributed algorithm that can construct a spanner in O(log n log Γ) timeslots with a high probability, where n is the total number of nodes and Γ describes the ratio of the maximum distance to the minimum distance between nodes. The constructed spanner concurrently satisfies two most desirable properties: constant stretch and linear sparseness. Our algorithm employs a novel maximal independent set (MIS) procedure as a subroutine, which is crucial in achieving the time efficiency of spanner construction. The MIS algorithm improves the best known result of O(log² n) [33] to O(log n) and is of independent interest as the algorithm is applicable also to many other applications. We conduct simulations to verify the proposed spanner construction algorithm, and the results show that our algorithm also performs well in realistic environments.