A new N = 4 superconformal algebra (SCA) is presented. Its internal affine Lie algebra is based on the seven-dimensional Lie algebra su(2) ⊕ g, where g should be identified with a four-dimensional non-reductive Lie algebra. Thus, it is the first known example of what we choose to call a non-reductive SCA. It contains a total of 16 generators and is obtained by a non-trivial Inönü-Wigner contraction of the well-known large N = 4 SCA. The recently discovered asymmetric N = 4 SCA is a subalgebra of this new SCA. Finally, the possible affine extensions of the non-reductive Lie algebra g are classified. The two-form governing the extension appearing in the SCA differs from the ordinary Cartan-Killing form.