Is it unlawful for me to buy my colleague’s car after a merchant has paid her a down payment, knowing that both of them are aware that she wants to wait several months, and is it permissible for me to offer the merchant a higher price?

It is forbidden to bid over your brother’s bid if the buyer and seller have agreed and settled on the sale, even if the transaction has not yet been completed. An exception to this is if the seller permits it, or if the buyer abandons the purchase, or if the seller abandons the sale to him. The buyer’s lying is not a justification for bidding over his bid.