In this paper we study Hankel operators in the quaternionic setting. In particular we prove that they can be exploited to measure the L ∞ distance of a slice L ∞ function (i.e. an essentially bounded function on the quaternionic sphere ∂B which is affine with respect to quaternionic imaginary units) from the space of bounded slice regular functions (i.e. bounded quaternionic power series on the quaternionic unit ball B). Among the difficulties arising from the non-commutative context there is the lack of a good factorization result for slice regular functions in the Hardy space H 1 . KEY WORDS AND PHRASES: functions of a quaternionic variable; Hankel operators.