Is it possible to design a deep CNN model on a small size image dataset
According to your description of the data, it is highly probable that training any neural network of reasonable size will overfit the training data.

One option is to apply any possible regularization approach; these come to mind:


Use L1/L2 weight regularization.
Reduce the number of parameters as much as you can (less layers, less number of filters).
Use dropout.


Also, try any data augmentation trick: rotate images, mirror images, crop and scale images, etc.