If we can consider electron as a small sphere then the question arises that if its spinning, it must radiate but its centre of mass is moving with a constant velocity, but an electron moving with constant velocity do not radiate. So will it radiate?
I prefer to rephrase your question as to whether a spinning ball holding a charge radiates electromagnetic waves or not. I mean forget about electrons, they are quantum pointlike particles! The answer is yes. First the charge could be either throughout its volume or only on the surface, or both. Then consider an infinitesimal volume and/or an infinitesimal element of surface: it is a charged body in a circular motion, so we are in the well-known case of the so-called synchrotron radiation, or magnetic bremstrahlung as Landau-Lifshitz call it (§74). As a result, the spinning ball would loose energy and eventually stop spinning (barring complicated details of its structure).
I'd like to add some things about spin in order to complete the good answer of Luc J Bourhis. It is pretty tricky to consider an electron as a spinning small sphere as you do. Spinning around what? You cannot properly define an axis of rotation in quantum mechanics. If you want a visual representation of a spin, it is a quantity that describes how the particle is when you turn around it. Imagine an arrow $\uparrow$ , if you look at it from the top and you turn around it, you need to make a 360°-turn to see exactly the same pattern again, which would correspond to a spin-1 particle. You need 720° in the case of an electron (spin one half).
Moreover, if you want to do an analogy between a spinning charged particle and a spinning macroscopic object, an electron would have to move faster than the speed of light to produce the magnetic moment that is measured experimentally.