Images in p tags?

Yes, it is OK, if the text equivalent (the value of the alt attribute) makes sense at that place in the paragraph.

Example 1

<p>If there is a known issue with the service, you will see a
<img src="/img/warning.png" alt="Warning!"> icon
in the upper right-hand corner.</p>

Example 2

<p><img src="/img/alert.png" alt="Notice! This service is temporarily out of order."></p>

If the text equivalent is empty, e.g., for a purely decorative image, it’s better to use a <div>.

Example 3

<div><img src="/img/sunset.jpg" alt=""></div>

Remember: an image is nothing but an alternative representation of the information in the text equivalent (semantically). Therefore an image can be used wherever the corresponding text equivalent can be used.