Given that, for the past $m$ days, the average daily production at a company was $100$ units per day.

- $1\;\text{day} \longrightarrow 100\;\text{units} $
- ${\color{Blue}{m\;\text{days} \longrightarrow 100m\;\text{units}}}$

If today’s production of $180$ units changes the average to $110$ units per day.

Now, $\dfrac{\underbrace{100m}_{{\color{Red}{m \;\text{days production}}}} + \underbrace{180}_{{\color{purple}{\text{Today’s production}}}}}{m+1} = 110$

$\Rightarrow 100m + 180 = 110m + 110$

$\Rightarrow 10m = 70$

$\Rightarrow {\color{Blue}{\boxed{m = 7}}}$

Correct Answer $:\text{C}$