This is a technical question:
Lang's Fundamentals of Differential Geometry (1999) has a proof of Frobenius' Theorem for $C^p$ ($2\le p<\infty$) on (possibly, infinite dimensional Banach) manifolds. To do this, he produces a chart, $\phi:U_0\times V_0 \to U\times V$ on pg. 159, where $\phi(x,y)=(x,\alpha(x,y)),$ and $\alpha(x,y)$ is essentially a component of a flow of a time-dependent vector field, which on a $C^p$ manifold is only $C^{p-1}$. Thus, a priori, $\phi$ is only $C^{p-1}$ and so could not be a chart for the $C^p$ manifold $X$. Can one actually show $\phi$ is $C^p$?
A similar issue arises in the "straightening out" proposition, 2.13 pg. 95, where the candidate chart is again obtained from a flow. Marsden, Ratiu, Abraham's Manifolds, Tensor Analysis and Applications has similar proofs of the two theorems and also do not address the degree of smoothness of the charts produced.
Lang alludes to this loss of differentiability on pg. 158, about the morphism $f$ and again in the global Frobenius theorem, pg. 164.) Again $p\ne\infty$ -- other proofs in other books work exclusively with $C^\infty$ (and finite dimensional) manifolds.
In more details:
On pg. 158, in the course of proving that an involutive distribution, $E$ (i.e. a tangent subbundle of $TX$) on a $C^p$ ($p\ge 2$) Banach manifold $X$ is integrable, he reduces the problem to one of local charts:
$$ 0\to U\times V\times\mathbb{E}\overset{\bar f}\to U\times V\times\mathbb{E}\times\mathbb{F} $$ where $\mathbb{E}$ and $\mathbb{F}$ are the model (Banach) spaces of the fibers of $TX$, and $\bar f(x,y,\mathbb{e})= (x,y,\mathbb{e},f(x,y)\mathbb{e})$, and $f:U\times V\to L(\mathbb{E},\mathbb{F})$ is $C^{p-1}$.
On pg. 159, he further reduces the problem to solving the PDE (or essentially the "classical Frobenius theorem, as mentioned by Warner in Foundations of Diff Manifolds....) $$ D_1\alpha(x,y)=f(x,\alpha(x,y)) $$
He solves this using proposition 2.1 on pg. 160, taking $\alpha(x,y)=\beta(1,x,y)$. Here $\beta$ is the solution of $$ D_1\beta(t,x,y) = f(tx,\beta(tx,y))\cdot x $$
He then takes a chart for manifold to be $\phi(x,y)=(x,\alpha(x,y))$, where $\phi:U_0\times V_0\to U\times V$.
From proposition 2.1 it seems that $\beta$ is a priori only $C^{p-1}$, and therefore likewise, $\alpha$, and $\phi$. Thus the supposed "submanifold" of $X$ given by $U_0\times y_0$ is only $C^{p-1}$.