If I understand your question correctly, you have already been ordered deported. If you case is still on appeal, you can request the Board of Immigration Appeals remand your case to the Immigration Judge to consider asylum, but only if there are changed conditions which have come up since the time of your hearing. Same is true if you did not appeal, but you will need a motion to reopen, rather than a motion to remand. Either way, you usually must have reasons for applying now that did not exist when you saw the Immigration Judge. But every case is different. You need to discuss your case with an attorney.
It may be possible. You may need to file a Motion to Reopen. Why did you not file for asylum during the original hearing? To have any chance you will really need to speak with and retain an immigration attorney.
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Legal disclaimer: The statement above is general in nature, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship.