Is there a wire (a bus), albeit a very short one, between the CPU and its level-1 cache?
No wires at least for the L1 and L2 caches, which nowadays are always on the CPU chip itself, and so they are connected to the rest of the circuitry with the same interconnects in silicon used for any integrated circuit, and are located as close to the cores as possible.  (Note: I am not considering the silicon interconnects within an IC to be wires, nor the traces on a PCB even though they function like wires.  Wikipedia doesn't either.)

L3 caches may be found either on the chip itself (as in the example below) or on the mother board.  In the latter case, where the connections to the from the CPU to the L3 is external, then these connections between the integrated circuit die and the pins on the chip are done with gold wires as shown below:

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So for a relatively short distance, yes there can be real wires involved.  But not for an L1 cache, which was the subject of your question.

The very first Intel processor to have an L1 cache was the 80486. It was 8K and located on the chip.  The L2 cache was 256K, and located on the motherboard.

Here is a picture of a four-core CPU:



The smaller ellipses are the L1 caches.  The larger ellipses are the L2 caches. They are each private to each core.  The large area is the L3 cache, which is shared between all of the cores.

Note that the caches take up nearly half of the area of the chip.

On my I7 CPU with four physical cores, there 4 x 64K L1 caches, 4 x 256K L2 caches, and one 8MB L3 cache.