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Let the number of shirts produced in the factory be $x$.

$15\%$ shirts are defective, then $85\%$ shirts will be non-defective.

- Defective shirts $=15\% \;\text{of}\; x =\frac{15}{100}x = 0.15x$
- Non-defective shirts $=\frac{85}{100}x = 0.85x$

$20\%$ of the rest are sold in the domestic market, then $80\%$ will be remaining

Now, $0.85 x \times 80\%=8840$

$\Rightarrow \frac{85}{100}x \times \frac{80}{100}=8840$

$\Rightarrow 85x \times 8 = 8840000 $

$\Rightarrow x = \frac{8840000}{85 \times 8} $

$\Rightarrow\boxed{x=13000}$

$\therefore$ The number of shirts produced in the factory is $13000$.

$\textbf{Short Method:}$

Let the number of shirts produced in the factory be $100$.

- Defective shirts $=15$
- Non-defective shirts $=85$

$20\%$ of $85$ sold in the domestic market.

- Shirts sold in the domestic market $=85 \times \frac{20}{100}=17 $
- Shirts left for export $=85-17=68$

Now,

- $68 \longrightarrow 8840$
- $100 \longrightarrow \frac{8840}{68}\times 100$
- $100 \longrightarrow 130 \times 100=13000$

$\therefore$ The number of shirts produced in the factory is $13000$.

Correct Answer $:\text{B}$