A simple model of a living cell which undergoes processes of growth and dissolution is described as a free boundary problem for a system of two reaction-diffusion equations; the condition on the free boundary is of the Stefan type. The special case of radially symmetric cells was studied in earlier work. This paper is concerned with the existence of symmetry-breaking stationary solutions, i.e. with solutions which are not radially symmetric. It is proved, in the two-dimensional case, that there exist branches of non-radial stationary solutions bifurcating from radially symmetric solutions; indeed, for any mode l, l [ge] 2, there exists a unique bifurcation branch whose free boundary has the form r = Rl + [epsilon] cos l[theta] + [Sigma]n[ge]2 [epsilon]n[lambda]n ([theta]), | [epsilon] | small, with [lambda]n ([theta]) orthogonal to cos l[theta].