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We report the discovery of a resolved (0.9 arcsec) substellar companion to a member of the 1–5 Myr Taurus star-forming region. The host star (2M0437) is a single mid-M type (T eff ≈ 3100 K) dwarf with a position, space motion, and colour–magnitude that support Taurus membership, and possible affiliation with a ∼2.5-Myr-old subgroup. A comparison with stellar models suggests a 2–5 Myr age and a mass of 0.15–0.18M⊙. Although K2 detected quasi-periodic dimming from close-in circumstellar dust, the star lacks detectable excess infrared emission from a circumstellar disc and its H α emission is not commensurate with accretion. Astrometry based on 3 yr of AO imaging shows that the companion (2M0437b) is comoving, while photometry of two other sources at larger separation indicates that they are likely heavily reddened background stars. A comparison of the luminosity of 2M0437b with models suggests a mass of 3–5 MJUP, well below the deuterium burning limit, and an effective temperature of 1400–1500 K, characteristic of a late L spectral type. The H − K colour is redder than the typical L dwarf, but comparable to other directly detected young planets, e.g. those around HR 8799. The discovery of a super-Jupiter around a very young, very low-mass star challenges models of planet formation by either core accretion (which requires time) or disc instability (which requires mass). We also detected a second, comoving, widely separated (75 arcsec) object that appears to be a heavily extincted star. This is certainly a fellow member of this Taurus subgroup and statistically likely to be a bound companion. [ABSTRACT FROM AUTHOR]