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It is commonly assumed that in ultrastrong laser fields, when the strongfield parameter of the laser field $\xi$ is larger than one, the electronradiation is well described by the local constant field approximation (LCFA).We discuss the failure of this conjecture, considering radiation of anultrarelativistic electron interacting with strong counterpropagating laserwaves. A deviation from LCFA, in particular in the high-frequency domain, isshown to occur even at $\xi\gg 1$ because of the appearance of an additionalsmall time scale in the trajectory. Moreover, we identify a new class of LCFAviolation, when the radiation formation length becomes smaller than the one viaLCFA. It is characterized by a broad and smooth spectrum rather than anharmonic structure. A similar phenomenon is also demonstrated in the scenarioof an electron colliding with an ultrashort laser pulse. The relevance tolaser-plasma kinetic simulations is discussed.

New class of violation of local constant field approximation in intense crossed laser pulse scenarios