The feasibility of obtaining exact analytical results in the realm of QED in the presence of a background electromagnetic field is almost exclusively limited to a few tractable cases, where the Dirac equation in the corresponding background field can be solved analytically. This circumstance has restricted, in particular, the theoretical analysis of QED processes in intense laser fields to within the plane-wave approximation even at those high intensities, achievable experimentally only by tightly focusing the laser energy in space. Here, within the Wentzel-Kramers-Brillouin (WKB) or eikonal approximation, we construct analytically single-particle electron states in the presence of a background electromagnetic field of general space-time structure in the realistic assumption that the initial energy of the electron is the largest dynamical energy scale in the problem. The relatively compact expression of these states opens, in particular, the possibility of investigating analytically strong-field QED processes in the presence of spatially focused laser beams, which is of particular relevance in view of the upcoming experimental campaigns in this field. The predictions of QED have been confirmed with outstanding precision in numerous experiments. The impressive agreement between the theoretical and the experimental value of the electron (g−2)-factor is customarily quoted as a prominent example [1]. However, the experimental scrutiny of QED becomes much less thorough when processes are involved, occurring in the presence of a strong background electromagnetic field, i.e. of the order of F cr = m 2 c 3 / |e| = 1.3 × 10 16 V/cm = 4.4 × 10 13 G (here m and e < 0 are the electron mass and charge, respectively) [2]. The main reason is that these values largely exceed the field strengths available in laboratories. An important exception is represented by the electric field of highly-charged ions (charge number Z ∼ 1/α, with α = e 2 / c ≈ 1/137) at the typical QED length λ C = /mc = 3.9 × 10 −11 cm [3,4]. Indeed, numerous experiments on processes occurring in the presence of highly-charged ions [5][6][7][8] have already successfully confirmed the predictions of QED. Correspondingly, advanced analytical methods [9], have been developed to interpret accurate experimental data beyond the exactlysolvable Coulomb model of the ionic field.Modern high-power lasers represent an alternative source of intense electromagnetic fields structurally thoroughly different from atomic fields. Although the amplitude F 0 of the strongest laser pulse ever produced is about 10 −4 F cr [10], it can be boosted to an effective strength F * 0 ∼ F cr in the rest-frame of ultrarelativistic particles colliding with the laser beam [11]. This principle has been exploited at SLAC to perform the so-far unique experimental campaign on strong-laser field QED [12], employing a laser with photon energy 1 eV and amplitude F 0 = 2.7 × 10 10 V/cm, and an almost counter-propagating electron beam with energy of 45 GeV (F * 0 ≈ 0.3 F cr ). The relatively large pul...