You should try and work it out with the buyers. Remember in order for the buyers to get a loan it must apprase and in this case it did not.

I agree with the other agent, but in the existing conditions think twice before loosing a buyer. Negotiate with the buyer and then always have a plan B. If you have enough equity in your home and not losing money then think about the bank appraisal as a guideline. Good luck, I know it is a tough decision.

No, you do not have to sell at the appraised value. If your contract has an appraisal contingency the buyer does not have to purchase your home either. You can neogitate the difference with the buyer, but unless it is a cash deal most lenders will not approve a loan when the sales price is higher than the appraised value. The only other option you have is to withdraw from the contract.

Good luck, I hope you and the buyer can resolve this issue.

Most sales contracts are subject to appraisal . However, most contract do not force the seller to sell at appraise value and do not obligate the buyer to buy at the sale contract .This issue became negotiable.
In today market appraising a house become almost a challenge. I remember in Miami six year's ago were homes sold NOT SUBJECT TO APPRAISAL . Buyers were buying homes above appraise value.
Finally , your contract become again negotiable .
Your option is either you sell at appraise value or withdraw from the contract or You can negotiate the sum between the appraise value and contract price.
Good Luck