Schwarz theorem on manifolds

Suppose $M$ is a manifold of class $C^2$ and I have a function $f:Mtimes M to M$ which lies in $C^2$. I can define a partial derivative such that I differentiate by one argument, e.g. if $varphi: mathbb{R}to M$ such that $X=[varphi]in T_pM$, then the derivative w.r.t. the first argument is $dfrvert_p(cdot,x)(X) = [f(varphi, x)]$. My question is if something like the Schwarz theorem holds, i.e. do the partial derivatives at a point $(p,p)$ commute? Under which conditions?

My thoughts so far were: Since $f$ is $C^2$, for any maps $varphi:tilde{U}to U$, $psi:tilde{V}to V$ with neighborhoods $tilde{V}subset mathbb{R}^n$, $tilde{U}subset mathbb{R}^{2n}$ and $Usubset Mtimes M$, $Vsubset M$ such that $pin U$, $f(p)in V$ and $f(U)subset V$ the map
$$psi^{-1}circ fcircvarphi: tilde{U}to tilde{V}$$
is in $C^2$ on subsets from $mathbb{R}^{2n}$ to $mathbb{R}^n$. Hence, the Schwarz theorem is applicable for each component function.

I came across this problem on the proof for the anti-symmetrie of the Lie bracket (Duistermaat), where $f$ is the commutator of the Lie group. There, we have the property that the derivative vanishes which I suspect is a necessary requirement.