Turns out I was wrong.
ArcGIS can import a .ter file but it can not export a .ter file.
For those that might be interested in this later here is the full list of import and export options currently within ArcGIS.
http://desktop.arcgis.com/en/arcmap/10.3/manage-data/raster-and-images/supported-raster-dataset-file-formats.htmIt also turns out that .png and .jpg can't be exported out of ArcGIS at 16 or 32 bit.
When looking more carefully through the documentation I noted that ArcGIS can export a 16 and 32 bit .tiff file so the person I'm working with at the company that is providing me with the files tried that. The base tiff file (8-bit) came in very blocky (their original data is at a 1 meter resolution) but when she exported it at 32-bit the file was big (365MB) but was completely blank... meaning it was pure white. There must be an error within ArcGIS but where that is occurring right now is unclear. I'm hoping she'll try and 16-bit and that will resolve it the problem tomorrow.
Oshyan, I'm unclear as to what the company is doing to the file before they give it to me but I know they not just using ArcGIS to export the file. I've asked for the original data and will see what I can do with it when it comes. I can't be the first one to do this so I assume there is an easy solution to the problem.
Some of the forums I'm reading are suggesting 32-bit can get too large and the program has a fit and gives up... hence the blank file.
I'll keep this post updated as I go through the process.