Replace the brake master cylinder.
Check for any signs of brake fluid loss (leakage) on brake system.
I would venture to say that it's the brake vacuum booster (between the master cylinder and firewall). Sometimes there's a small check valve in the line that supplies the brake booster with vacuum. Best case scenario It can go bad and cause just such a problem. Otherwise the diaphragm in the brake booster is probably going bad. I believe these are the starting point for a simple solution. However there may be other more costly options.
If there were a fluid leak the brake petal would most likely go to the floor with or without the engine running. The symptoms indicate to me that it's a vacuum problem as vacuum is only supplied while the engine is running and creating it.. However, it couldn't hurt to check for fluid deposits under the vehicle.
You can easily check by removing the vacuum line (rubber hose) from the brake booster. Depress the brake petal with the vehicle running and then off again. If it's symptoms remain the same: It's a vacuum issue.
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