We present a strongly interacting quadruple system associated with the K2 target EPIC 220204960. The K2 target itself is a Kp = 12.7-mag star at Teff ≃ 6100 K, which we designate as 'B-N' (blue northerly image). The host of the quadruple system, however, is a Kp ≃ 17-mag star with a composite M-star spectrum, which we designate as 'R-S' (red southerly image). With a 3.2-arcsec separation and similar radial velocities and photometric distances, 'B-N' is likely physically associated with 'R-S', making this a quintuple system, but that is incidental to our main claim of a strongly interacting quadruple system in 'R-S'. The two binaries in 'R-S' have orbital periods of 13.27 and 14.41 d, respectively, and each has an inclination angle of ≳89°. From our analysis of radial-velocity (RV) measurements, and of the photometric light curve, we conclude that all four stars are very similar with masses close to 0.4M⊙. Both of the binaries exhibit significant eclipse-timing variations where those of the primary and secondary eclipses 'diverge' by 0.05 d over the course of the 80-d observations. Via a systematic set of numerical simulations of quadruple systems consisting of two interacting binaries, we conclude that the outer orbital period is very likely to be between 300 and 500 d. If sufficient time is devoted to RV studies of this faint target, the outer orbit should be measurable within a year.