I am a bit late to the party, but came across this today as I had the same requirement (email with one image and image map for links) and thought I'd share what worked for me for any future onlookers.
I noticed if I created the email and then viewed with Internet explorer and clicked File->Send->Page by email then the links did not work in the resulting Outlook email. I use a third party to send emails from but by using IE it gives a quick preview of how it will appear in Outlook. The same markup when sent through the third party system did appear fine in Outlook and the image map links worked.
I then tried sending the same email programmatically with ASP.Net in C# and it also worked in Outlook. So I guess the problem may occur when creating the email directly from Outlook.
It would also be a good idea to add a simple link above the image opting to View in Browser and then link to the file on a server if possible.
The following link has another method that may work. I did not try this because the 3rd party or programmatically worked for me, but they explain that creating from Outlook will not work because Outlook is generating the email source code using mostly VML (Vector Markup Language)
http://www.outlook-apps.com/insert-html-to-outlook-emails/