First thing to do is describe the experiment in enough detail to think
about it.
What model probe master? link to specs?
3 "other" probes? More description might be helpful.
Divisions??? Frequencies are important. Sweep rate?
Not at all clear what you mean by separated by divisions and up and down
from center. Are you saying that the DC levels using the good probe
are not centered around ground but the bad probe is?
AC coupled looking?
What do you measure OHMS from the tip of the probe to ground with the
probe connected to the scope with the input set to DC?
What OHMS do you measure across the BNC with the tip shorted? Tip Open?
Can you get the BNC end apart to measure the cable resistance?
TEK scope probes had a resistive center conductor, but it wasn't
a huge number relative to the input impedance. probe master is unknown.
I'm betting that you have an open connection or a fried resistor
somewhere in the tip or compensation box.
If you know anybody with a TDR, they could make short work of it.
Check the 9Meg resistor in the tip. Depending on what's in the compensation
box or whether the probe has selectable 1X 10X, you might be able to
measure it without disassembly.
If you can disconnect the center conductor of the cable at both ends,
measure the capacitance at each end. If not the same, the ratio of
capacitances give some idea where the break might be.