Pico-8 demake of the classic arcade game Jungle King (by Taito 1982, shortly after renamed as Jungle Hunt, Tarzan replaced by an explorer)
Controls

?️ [Z] Action (Jump or Stab while underwater)
⬅️⬆️➡️⬇️ Movement, depending on stage:

Stage 1: none, just jump from vine to vine
Stage 2: swim in all directions while underwater
Stage 3: ⬅️ run faster, ➡️ slow down (they can be used during jump) and ⬇️ to duck
Stage 4: ⬅️ walk left, ➡️ walk right (no influence on jump)

The Game
It is divided into four different stages:

jungle and swinging vines (with monkeys on harder levels)
swimming in a crocodilesalligators-infested river
running uphill with falling rocks and boulders
a camp with cannibals where our hero's girlfriend dangles above a boiling cauldron (an extra native will throw his spear from the tree on harder levels).

Keep an eye on the TIMER: it starts at 5000, quite generous but it speeds up in the last stage and if it reaches zero it's an instant game over, it means your girl ended up in the cauldron.
Completion bonus and Extra lives
So yes, your objective is obviously to save the girl. In the congrats screen you will also see (an addition to the original game) time bonus points are accumulated until you reach 1000, this will give an Extra life! Time bonus is remaining timer 4. In the original Jungle King you will get 1000 pts regardless of timer which is a shame.
You also gain an Extra life every 30000 points.
Stage 1 - Swinging