Yes, you can. Click on the settings cog in the app, choose 'Downloads' and then make sure the 'Use mobile data' toggle is switched on. Downloading via mobile data will be switched off by default. If you can't see this option, make sure the app is up to date.
Downloaded programmes are available for the same amount of time as they are on the app. After the expiry, they will no longer be available. For more information, have a look here: How long am I able to watch a programme once it's been downloaded?
BBC iPlayer is a free service, but your mobile operator may charge for the data you use on their network. If you’re unsure how much mobile data costs or what your data allowance is, please contact your mobile network operator.