Is one obligated to fulfill a promise to sell to the first buyer, or is it permissible to withdraw from the sale for a neighbor who offers a higher price, especially since the first buyer did not pay a deposit?

As long as you have agreed on the price with the buyer and only the payment remains, this indicates that both parties are content with the sale. In this case, it is not permissible for your neighbor to offer you a higher price to buy your car, because this falls under forbidden bidding (al-sawm al-muharram), as the Prophet, peace and blessings be upon him, said: "A Muslim should not bid over the bidding of his brother." This is forbidden after the price has been settled and both parties have agreed.