Term for checking piece whose opponent daren't capture it What if any, is the term for a piece whose owner gets a draw by keeping up a relentless series of checks with it -- a paradox being that even if this player puts it en prise, the opponent daren't capture it because that would deliver stalemate?

The piece is in many cases a rook, and I have come across the terms "rambling rook" and "roving rook", but they seem to mean something else.

I've also come across the term "desperado", and that seems to mean something similar, but it is not the same. A desperado is a piece which might do some damage, e.g. delaying-checks, but they merely delay mate: those checks run out, and the other side captures the desperado without stalemating, and goes on to win.

The tactic is an unusual form of perpetual check. In most perpetual checks, the checking piece switches back between two squares on neither of which is it en prise.
I'd use the term you already mentioned, &quot;rambling rook&quot;, for this (at least when it's a rook). Tim Krabbé claims to have invented it:

If the term 'Rambling Rook' sounds unfamiliar, this could be because I invented it. In Russian it is beshenaya ladya, in Dutch dolle toren, both meaning 'crazy rook.' There is no English term, and I thought a little alliteration would be in its place.

Here are a couple of usages, showing that the term is quite widespread:
Rambling Rooks, compiled by SwitchingQuylthulg on chessgames.com

Rambling Rooks
Witness rooks selflessly throw theirselves at the opponent's mercy again and again... until there is no choice but to take them and accept draw by stalemate.

Can you stop the rambling rook? by user Sleafar on our sister site Puzzling Stack Exchange:

The rook can't be taken, because black couldn't move anymore which is a stalemate (= draw). If the king moves away the rook will continue to give checks.
Can you stop the rambling rook and find a way for white to win the game anyway?