First of all - congrats on the offer!

Most companies will pressurise you to provide a response asap, but you have the upper hand in this negotiation. They have chosen you, and now it is your turn to choose them. Although the stanstard for providing a response in the industry is 1-2 weeks, from my experience, it is doable to extend this by 1-1.5 weeks.

Having said that, you should be tactical about how you handle this:

1) Tell the other firms that you are interviewing with that you have an offer from a competing firm, in order to bring forwards your interviews and provide a response on existing rounds asap

2) Find out the deadline for providing a response to the firm that has given you an offer already

3) Ask them to extend the deadline, in order to think through the offer / finalise the interviewing process with the other firms... Although you can be honest, try to keep them warm. You can also delay providing a response, by setting up a meeting with HR/ a team member to go through any questions that you may have on the offer

Good luck!

Just tell them openly that you're in another recruiting process, which will probably take X weeks and you like to see what comes out of that. Maybe sweeten the cherry by saying your leaning towards their offer.

That's totally legitimate (nobody believes that they are the only company you applied to...) and they will probably be ok with it. After all, it's "just" an internship, you're not their CEO candidate. If they have a problem with it, they will tell you...

(edited)

Usually contracts specify when you have to make the decision. Didn't your contract have the date?