The first search for ultra-rare K+ decays into the pi(+)e(+)e(-) e(+) e(-) final state is reported, using a dataset collected by the NA62 experiment at CERN in 2017-2018. An upper limit of 1.4x10(-8) at 90% CL is obtained for the branching ratio of the K+ -> pi(+) e(+)e(-)e(+)e(-) decay, predicted in the Standard Model to be (7.2 +/- 0.7) x10(-11). Upper limits at 90% CL are obtained at the level of 10(-9) for the branching ratios of two prompt decay chains involving pair-production of hidden-sector mediators: K+. pi(+) aa, a -> e(+)e(-) and K+ -> pi(+) S, S -> A' A', A' -> e(+) e(-). (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP(3).