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 references 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