A topological space $X$ is defined to have *countable discrete cellularity* if each discrete family of open subsets of $X$ is at most countable.

A family $\mathcal F$ of subsets of a topological space $X$ is called *discrete* if each point $x\in X$ has a neighborhood $O_x\subset X$ that intersects at most one set $F\in\mathcal F$.

It is easy to see that a Tychonoff space $X$ has countable discrete cellularity if and only if for any continuous map $f:X\to M$ to a metric space $M$ the image $f(X)$ is separable.

A topological group $G$ is *$\omega$-narrow* if for any neighborhood $U$ of the unit there exists a countable subset $C\subset G$ such that $G=C\cdot U$. By a classical theorem of Guran, a topological group is $\omega$-narrow if and only if $G$ is topologically isomorphic to a subgroup of a Tychonoff product of metrizable separable topological groups.

It is easy to see that a topological group is $\omega$-narrow if it has countable discrete cellularity. What about the converse?

Problem.Does every $\omega$-narrow topological group have countable discrete cellularity?

A general version of this problem for uniform spaces has negative version.

Example.Let $X$ be an uncountable discrete topological space endowed with the uniformity, generated by the base consisting of entourages $$E_F=\{(x,y)\in X\times X:\{x,y\}\cap F\ne\emptyset\;\Rightarrow\; x=y\}$$ where $F$ runs over finite subsets of $X$.

It is clear that $X$ does not have countable discrete cellularily and the uniformity $\mathcal U$ is totally bounded.

Such a totally bounded example with uncountable discrete cellularity cannot be constructed in the framework of topological groups because of the following known

Fact.Each totally bounded topological group has countable (discrete) cellularity.

**Added in Edit.** Mikhail Tkachenko informed me that Problem has a counterexample (described in my answer below). However, the group in this counterexample is essentially non-complete.

Problem 2.Let $G$ be an $\omega$-narrow Raikov-complete (Abelian) topological group. Has $G$ countable discrete cellularity?

Also the following related question of Tkachenko remains open.

Question.Is there an $\omega$-narrow topological group containing continuum many pairwise disjoint open sets?