Sadly a lot of TOTP performances from the 1960s have been lost, but check out this fairly extensive list of TOTP performances from the 1960s that exist in the archive.
Dave Clark Five performing 'Bits And Pieces'
The Hollies performing 'Just One Look'
The Rolling Stones performing 'Not Fade Away'
Gene Pitney performing 'That Girl Belongs To Yesterday'
The Supremes performing 'Baby Love'
Roy Orbison performing 'Oh Pretty Woman'
The Moody Blues performing 'Go Now'
The Shadows performing 'Don't Make My Baby Blue'
Sonny & Cher performing 'I Got You Babe'
Jonathan King performing 'Everyone's Gone To The Moon'
The Byrds performing 'All I Really Want To Do'
Herman's Hermits performing 'Just A Little Bit Better'
Pink Floyd performing 'See Emily Play'
The Beatles performing 'All You Need Is Love'
Procol Harum performing 'A Whiter Shade Of Pale'
The Turtles performing 'She'd Rather Be With Me'
Dave Davies performing 'Death Of A Clown'
Alan Price Set performing 'The House That Jack Built'
Tom Jones performing 'I'll Never Fall In Love Again'
The Rolling Stones performing 'Let's Spend The Night Together'
Long John Baldry performing 'Let The Heartaches Begin'
Diana Ross & The Supremes performing 'Reflections'
Engelbert Humperdinck performing 'The Last Waltz'
The Bee Gees performing 'Massachusetts'
Dave Dee, Dozy, Beaky, Mick & Tich performing 'Zabadak!'
Cliff Richard performing 'All My Love'
Lulu performing 'Love Loves To Love Love'
The Foundations performing 'Back On My Feet Again'
The Move performing 'Fire Brigade'
Manfred Mann performing 'Mighty Quinn'
Status Quo performing 'Pictures Of Matchstick Men'
Herman's Hermits performing 'I Can Take Or Leave Your Loving'
Alan Price Set performing 'Don't Stop The Carnival'
Amen Corner performing 'Bend Me, Shape Me'
Brenton Wood performing 'Gimme Little Sign'
Dave Dee, Dozy, Beaky, Mick & Tich performing 'Legend Of Xanadu'
Manfred Mann performing 'My Name Is Jack'
Cilla Black performing 'Where Is Tomorrow'
Don Partridge performing 'Blue Eyes'
Julie Driscoll, Brian Auger & The Trinity performing 'This Wheel's On Fire'
Donald Peers performing 'Please Don't Go'
Peter Sarstedt performing 'Where Do You Go To (My Lovely)'
Cliff Richard performing 'Good Times (Better Times)'
The Hollies performing 'Sorry Suzanne'
Sandie Shaw performing 'Monsieur Dupont'
Marvin Gaye performing 'I Heard It Through The Grapevine'
Cilla Black performing 'Surround Yourself With Sorrow'
Love Affair performing 'One Road'
Lulu performing 'Boom Bang-a-bang'
Dave Dee, Dozy, Beaky, Mick & Tich performing 'Don Juan'
The Tymes performing 'People'
Stevie Wonder performing 'I Don't Know Why (I Love You)'
The Bee Gees performing 'The First Of May'
The Rolling Stones performing 'Honky Tonk Women'
Scaffold performing 'Lily The Pink'
Clodagh Rodgers performing 'Goodnight Midnight'
Marmalade performing 'Ob-la-di-ob-la-da'