  Fix a unital $C^*$-algebra $\mathscr{A}$, and write $\mathscr{A}_{sa}$ for
the set of self-adjoint elements of $\mathscr{A}$. Also, if
$f:\mathbb{R}\to\mathbb{C}$ is a continuous function, then write
$f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$ for the operator function
$a\mapsto f(a)$ defined via functional calculus. In this paper, we introduce
and study a space $NC^k(\mathbb{R})$ of $C^k$ functions
$f:\mathbb{R}\to\mathbb{C}$ such that, no matter the choice of $\mathscr{A}$,
the operator function $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$ is
$k$-times continuously Fr\'echet differentiable. In other words, if $f\in
NC^k(\mathbb{R})$, then $f$ "lifts" to a $C^k$ map
$f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$, for any (possibly
noncommutative) unital $C^*$-algebra $\mathscr{A}$. For this reason, we call
$NC^k(\mathbb{R})$ the space of noncommutative $C^k$ functions. Our proof that
$f_\mathscr{A}\in C^k(\mathscr{A}_{sa};\mathscr{A})$, which requires only
knowledge of the Fr\'echet derivatives of polynomials and operator norm
estimates for "multiple operator integrals" (MOIs), is more elementary than the
standard approach; nevertheless, $NC^k(\mathbb{R})$ contains all functions for
which comparable results are known. Specifically, we prove that
$NC^k(\mathbb{R})$ contains the homogeneous Besov space
$\dot{B}_1^{k,\infty}(\mathbb{R})$ and the H\"older space
$C_{loc}^{k,\varepsilon}(\mathbb{R})$. We highlight, however, that the results
in this paper are the first of their type to be proven for arbitrary unital
$C^*$-algebras, and that the extension to such a general setting makes use of
the author's recent resolution of certain "separability issues" with the
definition of MOIs. Finally, we prove by exhibiting specific examples that
$W_k(\mathbb{R})_{loc}\subsetneq NC^k(\mathbb{R})\subsetneq C^k(\mathbb{R})$,
where $W_k(\mathbb{R})_{loc}$ is the "localized" $k^{th}$ Wiener space.
