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Sage does have p-adic fields, but no-one has yet implemented completions of number fields at finite places (see K.completeion? where K is a number field). You should still be able to manually construct the apprpriate p-adic field and base-change the elliptic curve to it.
Here's an example with a split prime, which is easier:
sage: K.<a> = NumberField(x^2 - x + 1)
sage: Q7 = Qp(7, prec=20)
sage: e1, e2 = K.embeddings(Q7)
sage: E = EllipticCurve('389a1').change_ring(K)
sage: Q = E.gens()[0]; Q
(-1 : 1 : 1)
sage: Q.division_points(5)
[]
sage: E.change_ring(e1)(Q).division_points(5)
[(2*7 + 5*7^2 + 3*7^4 + 6*7^5 + 3*7^6 + 4*7^7 + 7^8 + 7^9 + 4*7^10 + 3*7^11 + 4*7^12 + 3*7^13 + 7^14 + 5*7^16 + 4*7^18 + 5*7^19 + O(7^20) : 6 + 3*7 + 7^2 + 4*7^3 + 5*7^5 + 5*7^7 + 2*7^8 + 6*7^9 + 6*7^10 + 3*7^11 + 5*7^12 + 6*7^14 + 5*7^15 + 4*7^16 + 7^17 + 6*7^18 + O(7^20) : 1 + O(7^20))]
