Convert the number in base nine to base ten.

The number $35_{\text{nine}}$ can be converted to base ten using the formula:

$$35_{\text{nine}} = (3 \times 9^1) + (5 \times 9^0)$$ $$35_{\text{nine}} = (3 \times 9) + (5 \times 1)$$ $$35_{\text{nine}} = 27 + 5$$ $$35_{\text{nine}} = 32_{\text{ten}}$$

The given equation is $44p = 35_{\text{nine}}$. Substitute the base ten value for $35_{\text{nine}}$:

$$44p = 32$$

To find $p$, divide both sides of the equation by 44:

$$p = \frac{32}{44}$$

Both the numerator (32) and the denominator (44) are divisible by 4.

$$p = \frac{32 \div 4}{44 \div 4}$$ $$p = \frac{8}{11}$$

The value of $p$ is $\boxed{\frac{8}{11}}$.