I’d like to share some files with one of my clients. Can I do this with SharePoint without having to pay for an extra Office 365 license for them?

Your Office 365 subscription includes Partner Access Licenses that you can use to allow anyone outside of your organization to access SharePoint.  This is called external sharing. 

External sharing is turned on for your entire SharePoint instance by default, but that setting can be changed by an Office 365 administrator.  When external sharing is turned on, there is quite a bit of flexibility in how it works.  An administrator can turn external sharing on or off for individual site collections, designate whether or not external users need to log in with a Microsoft account or can access the files anonymously, and determine which members of your team have the ability to send the sharing invitations to external users.  In addition, there are a variety of ways you can segment the files and other content that you would like to share externally from all your internal files.   

With all of these options, it’s beneficial to do a bit of planning and preparation before you begin taking advantage of the external sharing feature.  Microsoft provides a very nice overview of SharePoint external sharing here for Office 365 administrators.