Student or Learner
I know that the right way to say would be
We have 5 lessons a day.
I wonder whether "in a day" is always wrong in this context?
We have 5 lessons in a day. (meaning every day)
"We have five lessons a day" suggests regularity. An argument could be made for using "in a day" if it was a one-off and I would expect to see it in the past tense.
"I went to Turin for a week to learn Italian. I'm used to having three lessons a week here but during that trip to Turin, we had five lessons in a day!" You would also hear "... five lessons in one day!" in that context, with the stress on "one day!"
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.