The purchasing power of money (i.e., the inverse of prices) is determined by the stock of money and the demand for money. The stock of money consists of money proper and money substitutes. In a fractional reserve banking system, banks issue fiduciary media, i.e., money substitutes for which they hold only a fraction of money in reserve for redemption. It follows that the banking system can expand the money stock by issuing fiduciary media and thereby, generate price inflation.
Your friend has not thought through the logic of his example. If the 1st bank keeps $100 dollars and lends out the other $900 of the original $1,000 in cash deposited, then the borrower spends the $900 and the merchant receiving the money deposits it in his bank. The 2nd bank then keeps $90 and lends $810 (now the banking system has created $1,900 of money substitutes already on the $1,000 in cash). And so on the process continues until the banking system creates $10,000 in fiduciary media on top of the $1,000 cash reserve. So even if the original depositor doesn’t spend his $1,000 deposit, the fractional-reserve banking system is inflationary. Of course, instead of going through this indirect process, it’s more likely that the 1st bank simple issues a loan of $10,000 to a customer and puts the funds in his account in which case it would be meeting the reserve requirement ratio of 0.10 and yet it has created $9,000 additional dollars of money substitutes.