When is the projection $\mathrm{P}:x\longmapsto \mathrm{J}(x)^\top(\mathrm{J}(x)\mathrm{J}(x)^\top)^{-1}\mathrm{J}(x)$ Lipschitz continuous? 
*

*$f:\mathbb{R}^n\to\mathbb{R}^m$ is a smooth function with $n > m$.

*$\mathrm{J}(x)$ is the Jacobian matrix of $f$ and it is full row-rank at every $x\in\mathbb{R}^n$.

*For any $x\in\mathbb{R}^n$ define the projection matrix as the $n\times n$ symmetric idempotent matrix
$$
\mathrm{P}(x) = \mathrm{J}(x)^\top(\mathrm{J}(x)\mathrm{J}(x)^\top)^{-1}\mathrm{J}(x).
$$

Can I say that $\mathrm{P}: x\longmapsto \mathrm{P}(x)$ is Lipschitz continuous? If not, what conditions do I need to say this?

 A: In general the map $P:x\mapsto P(x)$ is not globally Lipschitz continuous, as @David E Speyer showed in the comments.
Since you are interested in sufficient conditions, here is a set. As usual with sufficient conditions, they are a little strong.
Theorem
Let $J$ be such that

*

*$J(x)$ is bounded by $C$

*$J^\dagger(x)$ is bounded by $C_\dagger$

*$x\mapsto J(x)$ is Lipschitz with $Lip_J$

*$x\mapsto J^\dagger(x)$ is Lipschitz with $Lip_{J^\dagger}$
where $J^\dagger(x) = J^\intercal(x)\bigg(J(x)J^\intercal(x)\bigg)^{-1}$. Then
$$
  Lip_P \leq Lip_{J^\dagger}C+Lip_{J}C_{\dagger}.
$$
Proof
This follows from
$$
   \|P(x)-P(y)\| = \|J^\dagger(x)J(x)-J^\dagger(y)J(y)\| \\ 
\leq \|J^\dagger(x)J(x)-J^\dagger(x)J(y)\|+\|J^\dagger(x)J(y)-J^\dagger(y)J(y)\| \\
\leq \|J^\dagger(x)\|\|J(x)-J(y)\|+\|J^\dagger(x)J(y)-J^\dagger(y)J(y)\|.
$$
The first term can be bounded using the boundedness assumption on $J^\dagger$ and the Lipschitz continuity of $J$ by
$$
\|J^\dagger(x)\|\|J(x)-J(y)\| \leq C_{\dagger}Lip_J\|x-y\|.
$$
The second term can be bounded by the remaining two conditions by
$$
\|J^\dagger(x)J(y)-J^\dagger(y)J(y)\| = \sup_{\|z\|=1}\|J^\dagger(x)J(y)z-J^\dagger(y)J(y)z\| \\ \leq \sup_{\|z\|=1}\|J^\dagger(x)-J^\dagger(y)\|\|J(y)z\|\
\\ \leq Lip_{J^\dagger}\sup_{\|z\|=1}\|J(y)z\|\|x-y\| \\
= Lip_{J^\dagger}\|J(y)\|\|x-y\|.
$$
Hence,
$$
\|P(x)-P(y)\| \leq (Lip_{J^\dagger}C+Lip_{J}C_{\dagger})\|x-y\|.
$$
QED
Observe that $cond(J(x)) = \|J(x)\|\|J^\dagger(x)\| \leq C_{\dagger}C$. This can help to reduce the strength of the assumptions on $J$ or $J^\dagger$.
