A = C^¥([0,1]), the algebra of all complex valued infinitely many times continuously differentiable functions on [0,1] is semisimple, for Rad(A) = Ç_{t Î [0,1]}{f Î C^¥([0,1]) ; f(t) = 0 } = 0. f ® f¢ is a derivation on A. The Johnson theorem says that 0 is the only derivation on a semisimple Banach algebra ( cf. [B&D, Theorem 18.21]). It follows that A = C^¥([0,1]) is not a Banach algebra under any norm.
For a proof based on the Singer-Wermer theorem see [Sak2, Corollary 2.2.4]). In addition a direct proof can be found in [Aup, Corollary 4.1.12].
This example is due to Silov([sil]).

Ref.

[Aup] B. Aupetit,A primer on spectral theory, Springer-Verlag, 1991.
[B&D] F.F. Bonsall, J.Duncan, complet normed algebras, Springer-Verlag, 1973.
[Sil] G.E. Silov, On a property of rings of functions, DoKl. AKad. Nauk. SSSR, 58(1974),985-8.