A ***bump function*** on a Banach space $X$ is a non-zero function with bounded support. Existence of a bump function of a given smooth regularity has, of course, strong immediate consequences --by translation and rescaling, one can make partitions of unity of that regularity, whence  regular approximations; and one can make ***smooth renorming*** of that regularity.  

A theorem of M. Fabian (see a reference below) states: *If $X$ possesses a $C^1$ bump function, it is Asplund* (i.e. separable subspaces have separable duals).
The simplest non Asplund space is $\ell_1$. Therefore, $\ell_1$ admits no $C^1$ bump functions. This I think was known even before, and there is a possibly simpler proof. (I suggest the above keywords in boldface for a search; some are listed below).  


M. Fabian, *On projectional resolution of identity on the duals of certain Banach spaces,* Bull. Austral. Math. Soc., 35 (1987), 363–371.

R. Deville, G. Godefroy, V. Zizler, 
*Smoothness and Renormings in Banach Spaces.*
Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64 (1993)

And this survey
https://core.ac.uk/download/pdf/81972936.pdf