If a word is in an English dictionary is that word necessarily an English word?

If a word is in an English dictionary is that word necessarily an English word? I'm in a discussion with someone who claims that not every word in the English dictionary is an English word, giving an example of 'sushi'. I know it's not originally English, that's what they call a loan word. My stance is if it's common usage It becomes a standard part of the vocabulary of the recipient language, and it will find is way into the dictionary. Please help me settle this. Thanks in advance.