We introduce a one-parameter family of transforms, U-(m)(t), t > 0, from the Hilbert space of Clifford algebra valued square integrable functions on the m-dimensional sphere, L-2(S-m, d sigma(m))circle times Cm+1, to the Hilbert spaces, ML2(Rm+1\{0}, d mu(t)), of solutions of the Euclidean Dirac equation on Rm+1\{0) which are square integrable with respect to appropriate measures, dint. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, U-(1) : L-2(S-1, d sigma(1)) -> HL2(C\{O}, d mu), to higher dimensional spheres in the context of Clifford analysis. In Clifford analysis it is natural to replace the analytic continuation from S-m to S-c(m) as in (Hall, 1994; Stenzel, 1999; Hall and Mitchell, 2002) by the Cauchy-Kowalewski extension from S-m to Rm+1\{0}. One then obtains a unitary isomorphism from an L-2-Hilbert space to a Hilbert space of solutions of the Dirac equation, that is to a Hilbert space of monogenic functions. (C) 2017 Elsevier B.V. All rights reserved.