Can I feed a subpanel protected by a 60 amp breaker with 6 AWG?

Yes, with a copper qualifier and a cable type qualifier.

Copper is fine, aluminum is not

Section 110.14(C)(1)(a) limits terminations on equipment rated 100 amperes or less to the 75 degrees C column of Table 310.16. In that column, 6 AWG copper is 65 amperes, which covers a 60 ampere overcurrent device with room to spare. 6 AWG aluminum in the same column is 50 amperes, which does not. For aluminum you need 4 AWG at 65 amperes.

Section 310.12 and its 83 percent allowance do not help here. That allowance applies to a dwelling service and to the main power feeder carrying the entire load of a dwelling unit. A subpanel feeder that serves part of the load is an ordinary feeder under 215.2(A)(1) and gets no reduction.

The cable type trap

If you are pulling 6 AWG THHN or THWN-2 in raceway, you are done. 65 amperes, 60 ampere breaker, no issue.

If you are running 6/3 NM-B, the answer changes. Section 334.80 requires the ampacity of NM cable to be that of the 60 degrees C column, and 6 AWG copper at 60 degrees C is 55 amperes. A 60 ampere breaker on a 55 ampere conductor looks like a violation of 240.4.

It is not, because of 240.4(B). Where the conductor ampacity does not correspond to a standard overcurrent device rating, the next higher standard rating is permitted provided three conditions hold: the conductors are not part of a branch circuit supplying more than one receptacle for cord and plug connected portable loads, the ampacity does not correspond to a standard rating in 240.6(A), and the next higher standard rating does not exceed 800 amperes. A subpanel feeder meets all three. 55 amperes is not a standard size, the next one up is 60, and 60 is well under 800. So 6/3 NM-B on a 60 ampere breaker is permitted.

The same logic applies to 6/3 SER used indoors, where 338.10(B)(4)(a) points you back to 334.80 and the 60 degrees C column when the cable is installed in thermal insulation. Outdoors and in free air the 75 degrees C column applies to SER.

What you cannot do is load that feeder to a full calculated 60 amperes and expect the cable to be comfortable. Section 220.40 governs the calculated load, and the calculated load has to be 55 amperes or less for the NM-B version.

The other three conductors

A subpanel feeder is four wires. Two ungrounded conductors, an insulated grounded conductor, and an equipment grounding conductor.

Table 250.122 sets the equipment grounding conductor at 10 AWG copper or 8 AWG aluminum for a 60 ampere device. The neutral is sized to the maximum unbalanced load per 220.61 and must be at least the size in Table 250.102(C)(1) if it is doing double duty, though in a subpanel it is purely a current carrying conductor.

Section 408.40 and 250.24(A)(5) require the neutral bar in the subpanel to be isolated from the enclosure. Remove or omit the main bonding jumper screw. Grounds go on a separate ground bar that is bonded to the can. Section 250.32(B)(1) prohibits using the grounded conductor as the fault return path to a separate structure and requires the equipment grounding conductor to run with the feeder.

Field takeaway

6 AWG copper on a 60 ampere breaker is correct. 6 AWG aluminum is not, use 4 AWG. With 6/3 NM-B you are leaning on 240.4(B) and the calculated load must stay at or below 55 amperes. Pull a 10 AWG copper equipment grounding conductor and float the neutral bar at the subpanel.