Let A be the free algebra on two symbols w,v, i.e. the algebra of all finite linear combinations of words in u and v. The set of all such words is countable, {w_n}, and we take the standard enumeration given by u,v,u²,uv,v²,u³,u²v,¼. Let B be the algebra of all infinite series x = å_{n = 1}^¥ a_nw_n, where ||x|| = å_{n = 1}^¥ ||a_n| < ¥. Then B is a non-commutative Banach algebra. Let x Î B, x ¹ 0, and let a_p be the first non-zero coefficient in the series å_{n = 1}^¥ a_nw_n. Then the coefficient of w_p^m in x^m is precisely a_p^m and so ||x^m|| ³ |a_p|^m  (m = 1,2,3,¼), r(x) ³ |a_p| > 0. Note that B is an infinite dimensional non-commutative Banach algebra in which the set of quasi-nilpotents coincides with the set of nilpotents.
Ref.
J. Duncan and A.W. Tullo, Finite dimensionality, nilpotents and quasi-nilpotents in Banach algebras, Proc. of the Edin. math. Soc., vol 19(Series II), Part 1, 1974.