Try not to think of Space and Time as 2 different things. Space and Time are ... Spacetime. Like the Wikiwiki says, "In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur."
Gravity is a force that affects objects in spacetime. Dense objects like planets, stars and black holes bend spacetime around them. Space is distorted around the object just like time is. Gravity affects time (spacetime) but its not a part of time. You can't separate them because they aren't together to begin with.
Many people do not believe that gravity has any affect on time and for sure, it's a far-out concept to grasp. But in reality it does, and everyone in the modern world already feels its effects.
GPS satellites are essentially flying clocks that broadcast their ID and its local time. The onboard computer adjusts for the bending of spacetime that the earth has on that satellite and its position in space. Without it, the onboard clocks would be off, and GPS navigation would be miles off, making them useless.
The warping of spacetime of the earth is caused partially by its mass, but also by something else called
frame dragging.
Airline pilots experience time more slowly when they fly. Of course it's a tiny fraction, but it can be measured. You can even say that they age more slowly, or that every time they fly, they arrive back on earth a few milliseconds in the future. Ack!