As the title says...
F3 dives and knocks a line drive down. He scrambles around on the ground for it and gets it in his bare hand but can't get to the base so he dives at the base and tags it with his glove while the ball is firmly in control in his bare hand.
Is your situation really any different than a fielder catching the ball while touching the base with his foot?
In a force, (or a force type situation), or an appeal situation, the base does NOT have to be touched with the ball or the hand holding the ball. The base just hast to be tagged, (with any part of the body), while the fielder has firm and secure possession of the ball.
For the record----
A legal attempted "out" tag is required with the ball secured in the fielder's tagging hand or in the fielder's tagging glove, applies when the play is being physically made on a runner.
An empty hand tag or glove tag will not get it done.
It will, as MD noted----when touching the applicable base.
2.00 TAG - a fielder who has the ball in hand or glove may tag a base with any part of the body - foot, other hand, glove, head (though I wouldn't recommend this last one).
Guess I'm "old" enough to have witnessed a "head force-out tag." Saw a "head tag" many years ago in a LL Majors LLWS District game.
Runner on 1st base, batter hit a bouncing ground ball to F6 who opted to make the force at 2nd himself.
In running to 2nd base, with ball in glove, F6 tripped, went sprawling on ground tantamount to a head-first slide toward the base. Glove hand with ball secured,
went to the left of the base; right hand went to the right of the base, neither one, or arms, touching the base. His head, minus his hat, touched the base before base-runner from 1st arrived.
Ump correctly called the base-runner "out."
Opposing manager ended up being ejected arguing the validity of the call.
Moral: live long enough as an avid baseball addict, and one "just might"--- repeat, "just might"----see or hear about baseball plays being made, never before made; or known to be made.
Needless to say, other sports as well.