Permissions at field level. Can this be done?
Not natively, but there's the Control Panel Body Classes plugin, which would enable you to add the handle for your "eventeers" User Group as a CSS classname to the &lt;body&gt; tag (i.e. body.usergroup-eventeers, if "eventeers" is the handle for your User Group).

With that classname in place, you could add some CSS to the Control Panel (easiest done w/ the CPCSS plugin) to hide the particular field, e.g. w/ something like the following:

body.usergroup-eventeers #fields-yourFieldHandle-field {
    display: none;
}


...or, if it makes sense, you could flip it around and have your field hidden by default, and visible only if the &lt;body&gt; has the .userlevel-admin classname - which would in effect hide the field from everybody except people w/ admin accounts:

#fields-yourFieldHandle-field {
    display: none;
}

body.userlevel-admin #fields-yourFieldHandle-field {
    display: block;
}


(Obviously, you'll need to change the yourFieldHandle part for the field's selector to your field's actual handle.)

In any case, keep in mind that this will not be a secure solution in the sense that the field will still be there, and any tech-savvy "eventeer" user can easily make the field appear by removing the above CSS rule in their browser's dev tools. However, in the current iteration of Craft it's as close as you'll come to accomplish what you want.