Does the category of local rings with residue field $F$ have an initial object? Let $F$ be a field. Does the category $C_F$ of local rings with residue field isomorphic to $F$ have an initial object?
This is, for instance, true if $F=\mathbb{F}_{p}$ for some prime $p$: If $R$ is a local ring with residue field $\mathbb{F}_{p}$, then any $x\in\mathbb{Z}\setminus(p)$ must map to something invertible under the morphism $\mathbb{Z}\longrightarrow R$. Hence that morphism factors as $\mathbb{Z}\longrightarrow\mathbb{Z}_{(p)}\longrightarrow R$; thus $\mathbb{Z}_{(p)}$ is the initial object.
But what happens in the more general case? I guess it should be true at least if $F$ is of finite type over $\mathbb{Z}$, but I have no idea how to prove it.
(EDIT - To avoid any confusion: I am talking about an initial object in the category of local rings $R$ with a fixed surjection $R\longrightarrow F$.)

Let $\mathbb{F_4}=\{0,1,w,1+w\}$ be the field of 4 elements.  Suppose $R$ is the initial object in the category described in the question for the field $\mathbb{F_4}$.  Then $R$ must contain some element $x$ which maps to $w\in\mathbb{F_4}$.  Thus we have a map $f\colon S\to R$, where $S=\mathbb{Z}[y]_M$, sending $y \mapsto x$.  Here $M$ is the maximal ideal of $\mathbb{Z}[y]$ containing $2,1+y+y^2$.
The following composition must be the identity: $$R \to S \stackrel f \to  R
$$
Thus  $R=S/I$ for some ideal $I\subset M$. Further we know $I\neq 0$ as $S$ cannot be the initial object: there are multiple distinct maps $S\to S$, such as the identity map and the map sending $y\mapsto y+2$.
Under the composition $S \stackrel f \to  R\to S$, we have $y\mapsto p/q$, for some $p,q$ integer polynomials in $y$. We know $p/q$ is not a rational number as $p/q\mapsto w\in\mathbb{F_4}$.  Thus $p/q$ is a non-constant rational function in one variable, taking infinitely many values, which cannot all satisfy the same polynomial over the integers.
On the other hand, as $I\neq 0$ there must be a polynomial over the integers satisfied by $p/q$.  This gives us the desired contradiction.
