You can be honest and refuse to disclose any information you don't want to offer.
They're trying to match their offer to the other company's, if one is made.
You made a mistake by identifying the other company. The proper response to the question which other company are you interviewing with is a peer company. You avoid identifying company A while implying they are equals, therefore a threat... which they can match.
Remember that interviews are negotiations, and negotiations are games; games where information is currency. Your opponent is better equipped and more experienced at this game than you. You don't want to give out too much information because not only will you get nothing for it, but it will hurt you by achieving a lower offer than you would otherwise.
Once you're done with your interview they will ask you if you got an offer, and will probably want to hear the details of that offer. Assuming you got one, the best play is to refuse to disclose any details. It puts the onus on them to try and match something they know nothing about, which can (and often does) lead to a higher number than you'd get otherwise.
If you don't know how to phrase this, I would go with something like
I've been asked to keep Company A's offer confidential. I cannot disclose any further details on that process, just as I wouldn't disclose any details of our interview process to them.
sidenote: this line also works brilliantly when considering a counter-offer from a soon-to-be ex employer
This is all well and good if you do get an offer from A, but what if you don't? Call Company B on the very next day and ask to resume the process.
Thank you for giving me the opportunity to interview with Company A before moving forward. I'm still interested in joining you, what is the next step?
If they ask whether you've got an offer, refer back to the other quote bubble. Stonewall them. It's none of their business. Let it be clear that any offers will be evaluated on their own merits and you want to focus on them, and what a brilliant fit you are for the role and the role for you.